It feels charmingly apt that when Tom Hanks – simultaneously one of the biggest movie stars in the world for the past 20 years and also famously one of most "aw, hell, shucks" normal men around – is talking about "the most fascinating days of my life", he is not referring to when he acted in zero gravity in Apollo 13, the time he became only the second actor in history to win back to back Oscars, or even learning to dance on a giant piano in Big. Rather, he is talking about his recent stint on jury duty.

"The people there, the process," he says, almost sentimentally recalling the days he spent trapped in a hot room with a dozen or so people "just eating my sandwich for lunch. Just fascinating!"

Sadly, Hanks's sandwich-eating came to an abrupt end because, being the highest-grossing film star on the planet, his presence attracted some attention and a mistrial was called when a member of the Los Angeles prosecutor's office approached him.

Hanks's life seems like one long battle to balance his high-profile persona with his determinedly low-key personality. A few days before we meet he appeared on the David Letterman Show in the US and mentioned that he has diabetes. Cue celebrity gossip meltdown.

"Yeah! I got type-two diabetes!" Hanks shouts out in the room, mimicking the media hysteria. "I'm sure there's going to be some media scandal now, saying I got it because I gained and lost weight for movie parts or something –" he gained two stone for his brilliant turn in A League of Their Own and lost three stone for the less fun Cast Away – "but I doubt that."

Many UK papers are claiming exactly that.

"Well, there you go. But it's just the way I've been living since I was 13. I feel a little embarrassed that everyone's asking about it now – I thought I was just saying the truth!"

Tom Hanks in Captain Phillips. Photograph: Smpsp/Hopper Stone, SMPSP

The reason Hanks is in London now, laughing good naturedly at the barrage of enquiries about his blood sugar, is because his latest film, Captain Phillips, is playing at the London film festival. In many ways, it is a typical Hanks role: a low-key family guy suddenly thrown into an extraordinary situation. It is based on a true story and Hanks plays Rich Phillips, a merchant mariner who, in 2009, was taken hostage by four Somali pirates. It's a brilliant film, as brain-bleedingly tense as you'd expect from a film directed by Paul Greengrass and Hanks is utterly fantastic in it, harrowing, heartbreaking and, as usual, deeply human.

This humanity is what Hanks looks for in a role: "Even if a story has nothing to do with my life, if I can recognise something of myself in the character and think, 'Oh yeah, that's what I'd do …' yeah, that's what I look for. That's the contract movies can pay off," he says.

Hanks is famously one of the nicest guys in Hollywood, and he is definitely nice, but he is also deceptively controlled, which may well have helped him to remain so nice. While he insists he is happy to talk "about absolutely anything", he lightly deflects any enquiries about his family, usually by pretending not to hear the question. He is also surprisingly fond of thespy cliches, using tedious terms such as "the process" and "getting there organically" when talking about his job, but I suspect this has less to do with pretension and more with obfuscation and that he is using luvvie chat to act as a smokescreen between himself and what he calls "the white heat of the spotlight".

In fact, with his chumminess but carefully maintained distance, talking with Hanks is like having an afterschool talk with a really nice teacher. Even when I admit that I haven't yet seen his 2012 film Cloud Atlas, he doesn't make some sarcastic remark about under-prepared journalists as many celebrities would. Rather, he gently chides me that I "really should", and then, teacher-like, returns to the topic later urging me to "check that film out. It's an amazing film." (The Guardian described it as "simplistic to the point of vapidity".)

Hanks in Big. Photograph: Allstar/20 CENTURY FOX/Sportsphoto Ltd./Allstar

Hanks was born and raised in California and had what he has described as a somewhat "fractured" childhood after his parents divorced. He discovered acting in school: "And I thought, this is much more fun than shop class! But," he Hanks-ishly adds, "shop can be good, too …" After college, he was cast in the TV show Bosom Buddies and caught the eye of Ron Howard, who cast him in his breakthrough role in Splash, a ridiculous but, thanks to Hanks, charming modern-day update on The Little Mermaid.

Even though Hanks achieved his greatest career success appearing in dramas in the 90s, such as Apollo 13 and winning his Oscars for (in typical Oscars style) his two worst films, Philadelphia and Forrest Gump, as well as appearing in romcoms (Sleepless in Seattle, You've Got Mail) written and directed by Nora Ephron ("She was a taskmaster, but gentle – I wish I was making a movie with Nora tomorrow"), I personally will always have a soft spot for his 80s comedic performances. He excels at combining sweetness with wackiness so that even if he's playing a man dealing with a collapsing house (The Money Pit), a smart aleck policeman (Dragnet) or a man in love with a fish (Splash), the part doesn't become subsumed in either slush or silliness. Even the most improbable of scenarios feel credible thanks to Hanks' unshakable normality.

The film that showcases this knack best is, of course, Big, the classic 1988 comedy about a 13-year-old boy who wishes that he was "big" and wakes up the next morning to find that he has turned into Tom Hanks. Hanks's performance is superb and he was deservedly nominated for an Oscar for it. But as much as I loved that film and still do, isn't it a little weird that in the film a child – in an adult's body, yes – has sex with an adult woman?

"Well, yeah, it is a little weird. But you know, the screenplay has it that it's a secret that he's a little boy. Elizabeth Perkins [who plays the adult love interest] thinks he's just some guy from work."

Yes, but he isn't.

Hanks with Geena Davis in A League of Their Own.

"Yeah, it was a little weird," he concludes.

Does he think they'd be able to make Big today, or would the whole "child has sex with woman" plot be a bit trickier to pull off?

"I think they could make it but they'd make it raunchier," replies Hanks, which is not quite the answer I expected. "That's what comedies are like today. I'm kinda amazed when I go see a comedy and it's just filthy – and with filthy language! I think: Really? Can they do this?" In true Hanks-style, he hastily adds a qualifier in case any of the guys from The Hangover get insulted: "But I gotta tell you, they're a guilty pleasure, those movies."

Is that why he doesn't really make comedies any more – because his comedy-style is more family-friendly than raunchy?

"The nature of live-action comedies has shifted without a doubt. Judd Apatow – his comedies are extremely different from the stuff I did, and that's just right and proper, the comedy zeitgeist has changed and I have to go with the parts that make sense for a guy in his late 50s."

I worry that this concern about age appropriateness means that he'll wince if I bring up one of the most 80s moments of comedy from that decade: his and Dan Aykroyd's rap duet, City of Crime, from their 1987 cop comedy, Dragnet, choreographed by Paula Abdul. But I risk it.

"Ho ho HO!" Hanks makes such a bark of laughter that it nearly hurls him out of the armchair. It is clear that my worry was unnecessary because he promptly launches into the rap: "See that, Streebek, we're just in time / We've stumbled into a major crime / They got all the frightened and that's not nice / I think she's the subject of a sacrifice!"

His technique has improved since Dragnet when he seemed to think "rapping" was another word for "randomly shouting". Now he remains utterly deadpan, even raising an occasional saucy eyebrow.

Has he been tempted to do any more rapping?

"Oh no, I don't think that's necessary. Watching that [rap scene] is like watching a fascinating home movie – I feel repulsed and fascinated at the same time," he says and, to be fair, that is probably how everyone else feels when they see Hanks and Aykroyd rapping in their too-tight shorts.

As it happens, this rap was the very first thing Hanks watched on YouTube: "The kids were little and I was talking about [the rap.] So the kids then went and pulled out the computer, plugged in the modem and they found it on YouTube. I said: Wow, if my kids can pull that up in two minutes, we are in a brave new world."

Hanks has been married to his second wife, Rita Wilson, for 25 years and the couple have two children, and he has two older children from a previous marriage. How did he manage to keep his family life so stable when his career went utterly stratospheric in the 90s?

"Well, I was lucky in that it didn't happen overnight and so I was used to this whole dog and pony show," he says gesturing his head with a 'gee shucks, me?' roll of his eyes around the grand hotel room. "And, you know, we had some of the greatest family vacations during that time – waterskiing in Seattle during Sleepless in Seattle, hanging out in Indiana during A League of Their Own …"

Hanks has frequently been compared to Jimmy Stewart and the similarities, as even he can admit after some self-deprecation, are obvious: "I get it – it's because he was unthreatening," he says. And, like Stewart, Hanks is simply so wholesome that it's impossible not to wonder what darkness lies beneath. Interestingly, Hanks's favourite Stewart movie is not the rather Hanks-ian It's a Wonderful Life, but Rear Window: "The logic of that film is impeccable!" he says happily. He'd make such an excellent villain, yet, with the exception of Cloud Atlas, he always plays the good guy.

"That's because I generally find villains illogical and if people can't look at a character and say: I'd do that, it takes them out of the story," he shrugs.

But does he regret missing out on any parts because they seemed illogical when he read the script?

"Oh no, I always make my peace with that. And if I go see a movie that I couldn't do for whatever reason I always come out thinking, oh no, I couldn't have done that."

For heavens' sake, Tom, this is ridiculous – how do you stay so gosh-darned nice?

He is silent for a second. "Well, you know, I understand the question," he says, not really looking like he understands the question at all. "But I've only ever been in this business for the fun. I only came here for the hang."

• Review: Captain Phillips

• Tom Hanks quiz