Something strange is going on. Carey Mulligan has requested me as her interviewer of choice. I'm not sure why, but this kind of thing doesn't happen often. If ever. One of Britain's brightest stars has insisted on unburdening herself to yours truly. We've never met. I'm baffled, but excited. Maybe she's got something she wants to get off her chest and sees me as the father confessor of celebrity journalism. I'll take that.

Mulligan, 28, doesn't give many interviews. Nor, for that matter, does she make many films. While in-demand actors churn out movies by the dozen, she is ever so picky about her parts. After being Oscar-nominated for her breakthrough film, An Education, in which she played a version of the young Lynn Barber, betrayed by an older man whose life was a lie, she took a year-long sabbatical because she couldn't find a role that satisfied her. She returned to work with some of cinema's best-known directors (Baz Luhrmann, Steve McQueen and now the Coen brothers) and leading men (Leonardo DiCaprio, Michael Fassbender, Ryan Gosling). Then she took another 18-month break while she waited for the next challenging role.

Mulligan likes to explore the darkest recesses: eternally disappointed Nina in Chekhov's The Seagull, a suicidal singer in McQueen's Shame, the downtrodden wife in Nicolas Winding Refn's Drive. Her characters are often quiet and restrained. Occasionally, too much so – her Daisy in The Great Gatsby is vapid. But she more than makes up for this in Inside Llewyn Davis, in which she plays a fabulously foul-mouthed folk singer, raging against her own pregnancy, men and the world.

Joel and Ethan Coen tell me they deliberately cast her against type. What is her normal type? "Sweet," they say simultaneously, in a phone call from New York. "People always seem to like her," Ethan says. "And for the actor that can sometimes be a burden. If you've got a part for a harpy, there's something appealing about casting somebody like Carey, and I think that's what appealed to her as well. She certainly took it up with relish."

What's she like as a person? "Sweet," they again chorus. "And shy," Joel adds. "But there's also a very no-nonsense side to her. She came in and got right down to business. That is one of those weird paradoxes you get with a lot of performers."

We agree to meet at a private members' club in west London. She leaves a message on my phone saying she'll be slightly late. Again, I'm surprised: stars normally leave the details to their publicist.

A few minutes later she arrives, a vision in brown – long brown hair rather than her more familiar peroxide pixie-cut, big brown eyes, brown jumper. She has a chameleon face; studious and sombre one moment, a dazzling elfin beauty the next. It's her perfect skin that is most striking: it's so luminous I can almost see my reflection in it.

As we wait for coffee, I ask what she wants to talk about. She tells me how much she has enjoyed working with the Danish film-maker Thomas Vinterberg on the forthcoming adaptation of Thomas Hardy's Far From The Madding Crowd, how delighted she was to be chosen by the Coen brothers for Inside Llewyn Davis, which she actually completed ages ago. It's a freezing day outside, cosy inside, the coffee is perfect and the world is good. Go on, I say. What do you think is most important for me to know about you? "I don't know," she protest-giggles. "Oh shit, nothing! Ask me a question – it's your job. I'm the actress. I'm just meant to sit here and wear a jumper."

Carey Mulligan with Justin Timberlake in the Coen brothers' Inside Llewyn Davis. Photograph: AP

Did she enjoy being cast against type? She bristles and asks what I mean. Well, I say, it was great to see you so sweary, giving it out rather than just taking it. She's not convinced by this thesis. So I mention that Kazuo Ishiguro once said Mulligan was perfectly cast as retiring Kathy H, in the movie of his novel Never Let Me Go, because she could play passive so brilliantly. "Yeahhh?" she says dubiously. "Certainly the character I played in Drive was passive." She's totting up her films in her head. "I don't think the character I played in Shame was passive. I don't think the character in An Education was passive, but maybe before that I was more on the back foot than the front foot." What she does bring to most parts is a magnificent stillness. Mulligan is becoming one of cinema's great observers: in Never Let Me Go, she's at her best watching over her rival in love, brooding, inviting us to read her mind.

Her early parts were largely in period dramas such as Pride & Prejudice, Northanger Abbey and Bleak House. "I used to have a typecast thing, and I fought my age and baby face. When I was younger, I used to play much younger than myself all the time and that got a little bit tiring. I don't think recently I've played a particular type."

It's interesting that she talks about fighting for her parts. Her most surprising role was in Shame, an uncompromising film about sex addiction, and boy, did she fight for that. She virtually begged McQueen. Why was she so desperate for the part?

She says it goes back to a career-defining piece of advice she got just after she had made Wall Street: Money Never Sleeps with Michael Douglas. "It was a great experience doing Wall Street, but it didn't feel there was a depth to the character. It didn't grip me in the way I wanted." She was in a rut, hadn't been offered anything she fancied, and her agent took her out for lunch. "She said you shouldn't do anything unless you can't bear the idea of anybody else doing it."

Sound advice, I say, but only one problem: if you don't find anything you're desperate to do, you end up taking early retirement. She nods. That's exactly what did happen. Actually, she says, the real problem was that she had been spoilt so early on in her career. I assume she means An Education and her Oscar nomination. Oh no, she says. What spoilt her was appearing on stage in The Seagull at the age of 21. "There's not many places to go for women after you've played Nina. She's the ultimate female role. I think I was looking to play her again in various incarnations." She tries to explain what makes Nina so special: despair, ambition, hope and ultimately redemption, all in the Russian girl who sees herself as the "soul of the world". She could talk about Nina for ever. "I felt a lot for her flaws. She was desperate to be loved and always reaching for something she couldn't get. That was the connection I made with Nina." The more she looked for a new Nina, the more despondent she became. Then she was sent the script for Shame, read the part of Sissy, and felt a similar empathy.

But Sissy is a desperate screw-up who appears to be burying an unmentionable secret from her childhood. Why does she feel such a connection with her? Mulligan shakes her head, unsure. Her parents are lovely, her childhood was privileged and largely happy, but there was just something. "I don't know… There was a tiny part of me… Like when I was 14 I went to school in Les Misérables T-shirts and my brother's jeans, and was always, like, not in the group. I wasn't on the outside, I had a great childhood, but I wasn't in the group."

She realised she was losing McQueen's attention when they first met, so she poured out everything – how she related to Nina, how no part had satisfied her since, how Sissy could be the one – then told him she was going to get a tattoo of a seagull as a permanent reminder. "He said, 'You're a fucking artist' and I said, 'Yeah!' Steve has this energy, he's like an American football coach, except he's British; he revs you up so you find yourself slamming a table in a coffee shop saying, 'I'm a fucking artist' before thinking, I've never spoken like that in my life. And he's like, 'And you fucking love it, don't you?' and I'm like, 'Yeah, I fucking love it.' And he's, 'Are you going to get the fucking tattoo?' and I'm, 'Yeah, I'm going to get the fucking tattoo.'"

Next morning she set off with her heavily inked friend Jenny to get tattooed. "I said, I can't go into a tattoo parlour in Soho, because I'd never been into a tattoo parlour in my life. I thought it would be full of bikers and be really intimidating, so we went to Selfridges and got my tattoo there." Why intimidating? "I don't know… People with tattoos are really cool! I thought I'd look too middle class and public school for it." She shows me the tattoo on her wrist. It's so understated, it's barely there.

Mulligan was born in London and had a peripatetic early childhood. Her father managed hotels all over the world, and when she was three the family moved to Germany. She and her brother attended the international school in Düsseldorf until the family returned to England when she was eight. It was at the international school that she first decided she wanted to act. "My brother was in The King And I, and I went to rehearsal and was completely enraptured by the whole thing, and desperately wanted to be in it. I was sobbing in the rehearsal room, so they let me be in it." Was she manipulative? She looks shocked. "No! I was earnest. That was the problem for most of my childhood – I was incredibly earnest. Just interested in one thing. I wanted to act."

With Leonardo DiCaprio in The Great Gatsby. Photograph: Sportsphoto Ltd/Allstar

Throughout her childhood, she says, she was a goody-goody – church-going, hard-working, obedient. She never did anything wrong because she was terrified of being caught. (She has still never tried drugs.) "I had a terrible guilty conscience from an early age. I don't know why." Was it anything to do with her Christianity? No, she says, religion was always a positive thing for her. From the ages of eight to 11, she went to convent school in England, and sang in the church choir throughout her childhood. "I had religion around me. It provided a real community and that lasted into my adulthood. Part of the reason I love doing films and plays is I love having a group of people you get to know really well; you're surrounded by people all the time. And I love that about going to church every week and singing in the choir."

Does she still sing in the choir? "No, but there's this woman called Mrs Jacobsen. She was my teacher at school and ran the church choir. Me and my best friend, Celia, went every week. When we were 15, we started helping her with the little kids. And whenever we're home for Christmas, we go back and do the church nativity thing with her."

Mulligan was a model pupil, found academic work easy, got eight A*s at GCSE. The first time she asserted herself, she says, was when she was 17 and told her parents she didn't want to go to university, she wanted to go to drama school and act. In the event, she didn't even go to drama school. How did her parents react? "They were like, what!" They thought it was a bad idea, a risky career made all the riskier by having nothing to fall back on. She had no contacts in the industry, so after seeing Kenneth Branagh in Henry V, she wrote to him, saying her parents didn't want her to act but she was sure it was her vocation. Branagh's sister wrote back saying, "Kenneth says that if you feel such a strong need to be an actress, you must be an actress." After screenwriter Julian Fellowes visited her public school, Woldingham, she wrote to him, telling him she wanted to be an actor. He invited her to dinner at upmarket restaurant Le Caprice, was impressed by her resolve, rang every casting director and agent he knew, and before she knew it she had a part in Pride & Prejudice alongside Keira Knightley.

How much does it help to be from a well-to-do family if you want to act? So many of today's stars have gone to public schools. Yes, she says, but there are also plenty who didn't – she mentions Fassbender and Knightley. "I think people who've been to drama school, it's certainly helpful to have money. I know lots can't afford to do it, but that's also not everybody's journey." As it happens, when she asked Fellowes for a tip, he suggested she marry a wealthy banker or lawyer.

She ignored his advice. Last year she married Marcus Mumford, of superstar folk band Mumford & Sons. There's a lovely story that Mulligan first met Mumford, whose parents established the evangelical Christian Vineyard Churches in the UK and Ireland, at a Christian holiday camp when they were children. I ask if this is true. And this is where the interview takes a turn for the weird.

"Erm… should we get another coffee?" she asks.

A change has come over her. She's twitchy and frosty. "I don't really want to talk about Marcus."

I point out that I'm bound to write about him, so I might as well get the details right. "You are so welcome to get things wrong," she says curtly.

"I'm interested that you were pen pals after first meeting as youngsters," I say.

"I know, but I don't want people to be interested in that. And your interest in it will inspire other people's interest in it, and more of my life I don't want people to know about will become exposed."

I'm flummoxed. OK, I say, I'll ask my questions and if you feel like answering any, do so.

With her husband, Marcus Mumford of Mumford & Sons. Photograph: Rex Features

"Did you keep in touch in the intervening years?"

"Shit. D'you know I haven't said no comment in a long time but it's just… no comment."

"That's fine," I say.

"Yeah. I don't want to be a pain in the arse about it, but that's just how I feel."

"You two live on a farm?"

"Yes."

"Do you farm land?

"Erm… No, we're not farmers."

Great, I say, you've answered something.

"I'm just saying we're not farmers, but that's all I'm going to say."

"Are there any animals? Much land?"

"All the stuff I don't want to answer."

"Have you got a dog?"

"Yes."

"What kind of dog?"

"He's a working cocker spaniel."

"What work does he do?"

She giggles. "He's got a paper round."

"Have you got a cow?"

"I have no cow."

Chickens? "No chickens."

Pigs? She gives me a look. I think I might have hit on something here.

"I'm just not going to answer," she says.

"You've got a pig, haven't you?" I say, excited at my potential scoop.

"No. Aargh. Don't. You really are so cheeky. I don't want to talk about it. We live in the country and that's literally my only… That's it."

"D'you want this coffee or not?" I say.

"I desperately want a coffee. I also desperately don't want to talk about Marcus any more. Or anything to do with the farm or life."

Ah, that might be problematic, I say.

"OK, you can ask, but I am now literally going to start saying no comment."

"OK. What was it like being married by Marcus's vicar father?"

"Pppphrr," she says as if struck by a chill wind. "No comment."

"How important is faith to you and Marcus as a glue?"

"No comment."

"Good question, though?"

"Great question. Wouldn't that be an interesting answer? No, I'm not talking about that."

"If you hadn't been pen pals, do you think you would have ended up marrying?"

"Resounding no comment. Great question. I must write these down for my autobiography when I'm 90."

What is her favourite Mumford & Sons song?

"Hehehe!" She's giggling quite manically now. "No comment!"

We sip at our coffees, unsure where to go from here.

I ask what would be suitable subjects to talk about. "The Coen brothers film I'm in and the Alzheimer's Society I work with; that's probably all that's interesting." And she does talk movingly about her grandmother's dementia, the fine work done by carers, how offended she was that people ridiculed Margaret Thatcher's dementia. "Jokes about her losing her marbles drove me crazy. It's such a throwaway term for someone who's literally losing their life in the most undignified, unpleasant way possible." If there was one thing she could change in the world, it would be the public perception of people with dementia. "The way people treat it, as if it's funny, really pisses me off."

Does she regard herself as political? "The area I'm most interested in is care for the elderly and Alzheimer's. That's the only passionate interest I have. I'm not particularly interested in politics."

Does she vote?

"Yes, I vote."

How?

"Privately. I believe in the secret ballot."

Back when she made An Education, Mulligan was compared to Audrey Hepburn and labelled the new It Girl. A journalist asked her what it felt like and she said, "Ask me in five years' time." Five years have now passed and I ask if she actually enjoyed that moment. "I didn't really have as much fun as I should have done. I found it all a bit terrifying. Now I don't get so nervous about standing on a red carpet and going to parties; I just get it done, whereas I used to be crippled by fear. Standing in front of a photographer was a nightmare. Now, as of last week, I've done a talkshow where I wasn't totally freaking out. My publicist said, 'This is the first time you haven't been completely terrified, and you were all right.' Usually I'm weird. I just get… I just get… I don't have a public persona. And when you go on those shows, you have to present a personality, you have to tell the anecdotes, whatever, and I can't really."

Her answer is so full, and so sad, that it helps me make sense of everything that's gone before – and her almost allergic reaction to media interest in her life. No wonder she always wanted to act when she was so uneasy in her own skin.

"Has your marriage to Marcus made you more secure?"

She smiles – a genuine, relaxed smile. "Yes, I think I'm more settled and more secure than I've probably ever been, and more confident and excited about the work choices that I make, so I feel my job isn't to be a dancing monkey and entertain people on television, it's to promote a piece of work I'm proud of. So I can't really fail if I stick to honestly representing my work and trying to avoid talking very much about myself. Which I seem to have done loads of today."

• Inside Llewyn Davis is out in the UK on Friday.

• The Coen brothers talk about the film

• Free audio download from the film

• Review: Inside Llewyn Davis