'I think we should all," Jon Hamm suggests, "get a chance to spend a week or two back in the shoes of our 24-year-old self…" Hamm, straight-jawed star of Mad Men, is talking in his rich bass voice about the premise of A Young Doctor's Notebook, an inspired adaptation of Mikhail Bulgakov's autobiographical tales of his apprenticeship as a doctor in rural Russia in 1916. In the drama, now entering its second series on Sky Arts, Hamm plays the middle-aged writer, looking back on the bleakly comic, steppe-back-in-time world of his youth, through the person of Daniel Radcliffe. The pair – the morphine-addicted 42-year-old dissident, and the naive 24-year-old medic – get to compare notes periodically, a set-up which among other things allowed the perfectly surreal spectacle at one point in the first, four-part series, of Don Draper sharing an antique copper bath with Harry Potter.

The adaptation was Hamm's idea, and he was instrumental in getting Radcliffe, who is a full foot shorter than him, as his innocent mini-me. With a passing facial resemblance, they make a likeable deadpan double act, one in which even the height difference feels right: "You think of your younger self as smaller and less worldly and more prone to making mistakes," Hamm says, "and anyway we are not dealing with realism here. You buy into it because neither can you talk to your older self in the bathtub." The harder challenge of the series lies in its delicate, mordant tone, which shifts quickly from shop of horrors medical emergency, to poignant reminiscence, to escapist slapstick. Think M*A*S*H meets Chekhov. Hamm is often faced with the challenge of conveying all those registers at once in his advice to the flailing, amputating, Radcliffe: "You just might have to settle for saving the world three-quarters of a peasant at a time," he will say.

Jon Hamm and Daniel Radcliffe in A Young Doctor's Notebook. Photograph: Sky Arts

"We didn't want it to be some dry historical drama," Hamm suggests, "but nor did we want to it to be a David Lynch artsy thing." We are talking after a day's shooting, in a deserted canteen at the show's studio in Twickenham. Hamm read, he says, a lot of Russian literature back when he was a student in Texas, "from Tolstoy to Solzhenitsyn", and came away from most of it thinking primarily: "Wow! Russia is a very different place from the midwest." You could, he says, just about imagine yourself there, but not that easily. He wanted the series to carry that alien sense, "a bit like how Americans feel about Downton Abbey".

He'd met Radcliffe at various awards ceremonies and parties. (It was a bonus that the Harry Potter star turned out to be an ardent fan of Bulgakov's The Master and Margarita.) "A couple of my friends had worked with Dan. And I knew he would spark to it," Hamm says. "He is incredibly dedicated, and not afraid to slip and fall and make a fool of himself, which is everything you want in an actor. He is off working on stage in the evenings while we are doing this [in The Cripple of Inishmaan at the Noël Coward theatre in London's West End] So: the perfect crazy 24-year-old."

The adaptation, is, among other things, I suggest, the ideal vehicle for the pair of them to be cast against their respective indelible types. Having spent six series as Don Draper, Hamm is beginning to imagine life without his impeccably tailored, semi-psychotic alter ego, who he will leave behind for good when series seven concludes in 2015. He doesn't see his current months off-set specifically as a holiday from Draper, he says, but he does go out of his way to find parts for himself where "he is not playing a philanderer" – though of course those are the parts that he mostly gets offered. "In that sense I am not unlike Mr Radcliffe. Our names tend to come with something attached – an audience and an expectation, and I guess a price tag. The challenge is to choose the right things to do."

The fascination with Hamm as an actor, one understood exactly by Mad Men creator Matthew Weiner, is that in all of his parts he appears to have arrived seamless and fully formed. The opening line of the first show of Mad Men asked the question "Who is Don Draper?" and those millions of "Maddicts" who have stuck with it are still obsessed by the answer.

Weiner imagined the Draper character as a Gatsby figure, the American archetype of the self-made hero with the tragic flaw. When he auditioned Hamm, he saw immediately what he had been looking for: an old-style Hollywood leading man, a Cary Grant or a Gregory Peck, an iconic adult male, but one who might convey both an intellect and a kind of emptiness, an emotional space to be filled. When Hamm left the room after his first audition, Weiner famously turned to his casting director and said, with certainty, "That man was not raised by his parents." Even now, off-duty, in a sweatshirt, and nursing a cold, he carries something of that quality about him; he is open and charming and quick to laugh, but also unfathomably self-contained in some way. If, I ask him at one point, he was in a bath with his own 24-year-old self, what would he be telling him?

Jon Hamm as Don Draper, with Jessica Pare as his wife, Megan. Photograph: AP/AMC

Hamm smiles at the idea. "I'd be telling him, just work harder. I had been doing plays back then. At 24, I'd just come to Los Angeles." He'd arrived in his beaten-up Toyota, with about 100 dollars stuffed into his jeans pocket, all his worldly goods. "When you think of yourself in your 20s you tend to obscure some of the worst memories and inflate some of the good parts," he says. "If I think back to my first couple of months as a professional actor, I cringe: I was such an idiot. In film and television there are so many customs that you don't know about. You just blunder in."

By 24, Hamm had done a great deal of enforced growing up. Born in St Louis, Missouri, he was two when his parents divorced; he lived with his mother and saw his father at the weekend. His mother had moved to St Louis from a small town in Kansas at 18 to work as a secretary. She met and married Dan Hamm, an older widower who already had two daughters. Hamm's father weighed more than 20 stone and was nicknamed "the whale" on account of his gargantuan personality as well as his girth. After his haulage business failed he sold cars, and tried his hand at advertising. When Hamm was 10 his mother died quite suddenly of stomach cancer. He was given a book, How to Deal with the Death of a Parent, and he moved into a house that his father shared with his own mother.

It was not a happy home, and Hamm escaped into sport – he was a serious baseball player – and increasingly into the theatre. He won a scholarship to Texas University. During his first term, his grandmother died, and during his second, on New Year's Day 1991, his father, who suffered from diabetes, also passed away. Hamm was 20, and, I imagine, suddenly profoundly rootless?

"Not rootless," he says, "but certainly not rooted. If you are in that situation you tend to cling on to people and cling on to things. For a few years I lived in a bunch of basements and on a bunch of couches, and had a lot of surrogate families around St Louis and Columbia, Missouri. A lot of those people are still in my life and that is a nice thing to have. It's maybe the occasional email – I don't do Facebook or Twitter or any of that shit – rather than a face-to-face meeting, but I keep up with people."

The cliche is that acting provides a family, and Hamm is not going to argue.

"We used to joke that the theatre department at college was where all the orphans ended up," he says. "They had been kicked out of every other place, but they found a role, a place there, whether it was front of house, or on the stage. That doesn't mean it wasn't bitchy and catty and competitive at times – like any family. But it took all comers."

What was the moment, I wonder, when he knew for sure that was where he belonged?

"It really started at high school," he suggests, "when one of my teachers, who is still a good friend, took me on one side and said: 'You are really good at this.' I still have problems with compliments. But that was the first one, and I kept hearing that as I carried on. I never really heard it in any other arena, no one ever said you are a really good writer, or you are good at math. And, as you say, I later found myself unrooted. I had no immediate family, no wife, no particular place to live, so I thought I might as well give this life a try. I've learned a few things, but one is: you are only 24 once. You have to hustle a bit to get where you want to be."

For a long while, living initially with his aunt and uncle in LA and later with friends, Hamm did the rounds of audition and rejection. He kept working – albeit at one point briefly as a set designer on a soft-porn film. It was not until 2000, at 29, that he got his first TV part, in a forgettable series called Providence; and when Mad Men came around in 2007, he was still scratching around for regular work, by now the other half of Jennifer Westfeld who had given him a role in the 2001 hit comedy Kissing Jessica Stein which she had written, and in which she starred. Given that he had to wait for it, his stellar Mad Men years must still feel a bit like the most surprising of plot twists?

Jon Hamm with his wife, Jennifer Westfeldt, who says: 'People come up and say all sorts of wildly inappropriate things… It never gets any less weird. It's like being with a Beatle or something.' Photograph: Broadimage/Rex

"It is. I mean, the stated career objective was just to work and pay off my college loans. Anything else would be gravy. I never wanted to be Tom Cruise or whatever. I just wanted to work on things I liked and to avoid douchebags. Thankfully that's how it worked out."

All men measure themselves against their father to some degree. When he started out as Don Draper Hamm would apparently keep a picture of his larger-than-life old man on his wall. In one early Mad Men interview he remembered looking at himself in the mirror the first time he put on one of Draper's trademark suits. "I was sitting in my dressing room," he recalled, "running lines in the mirror, and I thought: Oh, my God! I look just like my father. He was just so fucking sad. Here's this guy that looks like he's got the whole package, but there"s nothing inside but sadness."

Does he still feel that ever-present absence?

"I do," he says. "I mean I lost my dad when I was 20, and I realise now I never really had any what you might call adult conversations with him. There was a lot unspoken. I never heard how he met my mum for example. Or any stories about his childhood or much of what he had done before I was around. I had my own experience of him, but a kid's experience of watching his father is not necessarily the reality. I can see a resemblance in myself to my father, although of course I hope I keep my hair and don't gain 200lbs and don't get diabetes. He was an interesting guy, but a sad guy. He lost the two women he loved. It was pretty tough."

Watching Mad Men over the years it has always felt that Hamm's own story is around the edges of Don's life, is he conscious of that?

"Yes. I think it's pretty close to the surface. I'm not shy about that. I think Matt Weiner and the writing staff have wisely mined that source for its dramatic motherlode so to speak."

Does he access that history when he is acting in certain scenes?

"No, I generally don't," he says. "I'm not that methody. I am very cognisant that I am playing a character. Don Draper is a pretty dismal, despicable guy, so why I would want to take him home with me I don't know... It's a strange thing. People tell me they look up to Don, like they look up to Tony Soprano or Walter White [in Breaking Bad]. People have these weird fascinations with people who in reality you would not want to be for a second. There seems to be that vicarious thrill. Maybe it is the fact of doing everything wrong and getting away with it."

Perhaps because of his childhood awareness of how badly things can go wrong in life, perhaps because of his proximity to Don Draper, Hamm has led an apparently blameless Hollywood existence, even since his alpha male fame. I interviewed Jennifer Westfeldt, his partner of 15 years, at the time her latest film, Friends With Kids, came out, and she spoke then of the disconcerting reality of being out and about with Hamm in recent years. "The hardest bit for people," she said, "is that Don Draper is such an iconic character that fans of the show have a really hard time imagining Jon is not like that. That he is really this goofball. And that he is in this long relationship. That is, I guess, why people come up and say all sorts of wildly inappropriate things to him, and me, which don't bear repeating. It never gets any less weird. It's like being with a Beatle or something."

Hamm routinely fares well in sexiest/best-dressed male lists in both straight and gay publications. Along with his multitude of Emmys, he is indisputably GQ's man of the year, every year. Is that something he enjoys?

"Well, it is and it isn't a good thing. If you are a guy who needs that kind of validation, more power to you. I'm not a guy who needs that. I find most of it silly. You can ask anyone who knows me if I walk around as cock of the walk, and I'd be amazed if anyone said that was the case. That kind of culture should be the preserve of teenage girls, and if you ask any 16-year-old girl, she'll say I remind her of her dad. You know, I am a 42-year-old guy in a world in which Ryan Gosling exists…"

A comparative escape from some of that kind of attention is among the reasons Hamm enjoys working in Britain. He came here first as a chaperone for a theatre group from his old high school, his first trip abroad, aged 21. And though it was, he suggests, "a truly terrible idea to put me in charge of 20 13- to 17-year-olds, I anyhow fell in love with England, Stonehenge and Bath and Oxford and all that. I still love being here – the way it is based around walking and biking. I can wander around quite happily in London. I am kind of out of context. And unless it is late and they are shitfaced most Brits are too polite to hassle me."

Westfeldt is in New York, starring in a play. Hamm spent the summer in India, and in Atlanta, working on a Disney film about a baseball agent trying to hire cricket stars, so they have not seen much of each other. "It's the usual story," he says, a reality which means among other things they have never felt the time right to have children (the mostly comic subject of Westfeldt's last film, in which Hamm also starred). When I asked Westfeldt about her thoughts on a family she said: "I think kids should have something grounded. Jon and I are both children of divorce. I'm not sure we can do it without a proper home base." He agrees, suggesting St Louis would be a better place to raise children than either New York or LA, but that for them to move there would be like "a Texan oilman moving to Hawaii".

So it's not on the horizon? "It is not imminent, let's put it that way." And Hamm has no plans to slow down in his career any time soon. At one point in our conversation, we talk about actors' lives he admires.

"I have no idea how Tom Cruise, say, can still be credible as an action hero, but he is," he suggests. "That is crazy. I mean I have met him and he is a very intense guy: if there is one person on the planet who can carry that off for three decades it is probably him. But a guy like Jeff Bridges would be more my hero. He lives in Montecito, hangs out, and occasionally comes forward to do something great. I met him when he was nominated for an Academy Award for True Grit and I was totally starstruck, like a gibbering idiot, barely able to shake his hand."

While I imagine for a moment Hamm tongue-tied in a tuxedo, I wonder if he has given much thought to Don Draper's eventual demise. Having lived with him so long is it something he would like a say in?

"Well, it hasn't been written yet," he says. "But I have every possible faith in Matt [Weiner]. We have a meeting before every season starts and we sit and talk for an hour or two, not about specific things. Just chatting about what we did on holiday, what art we have seen, about sports or getting older or whatever. Just stuff that is on my mind. And he will write a bunch of things down from that conversation. I have no idea at all what the process is after that."

Having known intimately the loss of one family, I wonder how he will cope with losing the surrogates in Mad Men? Can he conceive of life without them?

"I hope so," he says. "It has been a long stretch. If you look back at Don and Betty in series one, and how young the kids were, and how much more hair I had, you get a real sense of life passing; it's a bit like one of those Russian novels in that way. It has been a solid 25 per cent of my existence on the planet. But that's enough, I think." Hamm smiles his broad Don Draper grin.

"Will it be sad? Absolutely. I have made a lot of very close friends on the show. You know, I am feeling pretty zen about it just now, but probably next year I will be crying my eyes out."

A Young Doctors' Notebook series 2 starts on Sky Arts 1 on Thursday, 9pm