When William Burroughs wanted to research his book Naked Lunch, he was in Paris. So he went straight to the Rue de Bucherie on the Left Bank, through the doors of the bookshop Shakespeare & Company, and directly to the bookshelves of the American bookseller George Whitman. There, he found stacks and stacks of hefty medical tomes, along with just about every English language paperback of note you could want. Burroughs has long gone from this world, but Shakespeare & Company has not. It is still perched on the same cobblestones on the bank of the Seine, overlooked by the Notre Dame. And if Willy Wonka were to take time off from chocolate concoctions and open a bookshop, this is what it would look like.

At Shakespeare & Co, there's not enough space for a stockroom, so it's a constant merry-go-round of books bought and books sold; tourists flock here to take photographs of the higgledy-piggledy interior, with books stacked from floor to ceiling. This is the place, after all, where in the Fifties the Beat poets hung out, and, more recently, where Ethan Hawke is filmed in the opening scenes of cult movie Before Sunset (and, indeed, where Meryl Streep in last year's Julia & Julia was seen to wander, in search of a cookbook). There's a wishing well in the floor that holds a plenteous supply of coins; in times gone by, it had a gas pipe which owner George Whitman was inclined to light on occasion (once, the story goes, when he was feeling particularly rakish, he accidentally set a hair-model's long tresses on fire). Upstairs, at the top of the winding staircase, there are all sorts of places for readers to loll. One room is a library, with literary donations to Whitman from Simone de Beauvoir's personal collection, and an eclectic selection of his own books – what remained, anyway, after a horrendous fire a number of years ago destroyed thousands of words. That's the room with a piano (and a fire extinguisher). When I visit, most customers aren't shy about playing it; although one man, too bashful to perform, is unable to resist sitting at the stool – he mimes playing for 10 minutes, fingers never touching the keys.

In the Fifties, Allen Ginsberg, William Burroughs, and a crowd of Beat poets made pilgrimages here, to the bohemian beacon of literature and licentiousness that promised a free home to travelling writers, aspiring or renowned. Whitman presided over his free-wheeling enterprise with an impressively prominent goatee, a critical eye, and a (mostly) open-door policy. From its inception, though, the labyrinthine bookshop was always more than just a place to buy a novel. It was a temporary home for writerly waifs and strays who tumbled in, slept among the bookshelves when the shop was closed, and wrote and wrote until they tumbled out again. George called them his "Tumbleweeds". He wanted people to have a place to stay when they came to Paris, to have a window on the world, a home from home in a strange city, even if they had no money in their pocket. His motto, "Give what you can, take what you need" was the creed of what became known as "Hotel Tumbleweed", a philosophy on life that was as radical then as now.

As a result, Shakespeare & Co has, over its lifetime, played host to hundreds of thousands of visitors. Some stayed a night, camping on the sofas in the library, or bunking down in the writer's studio, a little room with a typewriter and a curtainless window that at night lets in the light from Notre Dame. Others stayed months, even years (one legendary Tumbleweed reportedly stayed for seven).

George Whitman's single requirement was that by the end of their stay, a Tumbleweed must write their life story on a page, and attach a photograph of themselves. This policy still stands. George is 96 now – reading one book a week rather than the book a day he accrued throughout his tenure – and still receives every biography personally, usually from bed, in his apartment on the third floor of the bookshop. He tells me my biography is "pretty sloppy" and motions to the ripped page I used. "On an old scrap of paper? People usually do it with more style," he says, a little bit disgusted. "The Irish are usually good writers," he adds, leaving it unclear whether this is a compliment or an unfavourable comparison.

Age has certainly not dulled his critical sensibilities, and he's read plenty of biographies – the shelves that line his living room are testament to that. This is where the other Tumbleweed biographies are housed, hundreds and thousands of them. Some are hand-written, many type-written, the more recent stories computer-printed, all contained in neatly-bound folders marked by year, from the early Sixties right up to the present day.

During my short stay there, by accident I found myself locked in George's living room as he slept. Sharing the room with his dog and cat, and petrified of waking him, there was plenty of time for quiet reading. There is power in the collection of so many strangers' stories side by side. Every page is the distilled essence of a life, with thousands of hopes and dreams and aspirations crammed into each black folder. Margaret Gillerman, a fresh-faced student from Missouri, smiles out of her black-and-white passport photo. Born in 1951, she arrived on 7 July 1973 and left two days later. "As I left home... Dad cried out to Mom, 'There goes the tennis-shoed wonder'," she writes. "So I left that known and generous world and sprinted off on my tennis shoes. I hope to explore many, many winding and entwining paths so I can begin to understand what commonality exists between the many peoples and ways..." Iain Pickard, a "Lieutenant in British Army of The Rhine", arrived on 27 July 1980, and stayed until 2 August. He was 22. "My friends deflate me by calling me Pix, which is a good thing," he types. "Although happily embraced by the English Establishment, that does not mean that the hopes and idealism that are the indulgence of the young have passed me by... My philosophy is simple: there are things wrong with the system, deep and lasting change will only come from within... I have doubts. Soldiering as a profession is ethically distasteful, as a means of earning a living, however, it is constantly challenging and thus enjoyable."

The hope contained within is moving, so universal and out of time, that it is a shock when I remember that many of these people, then in their teens and twenties, are in their sixties or seventies now, some of them maybe even dead. Later I speak to Whitman's daughter, Sylvia, about this. "Yes, it was terrible when we tried to track the writers down and discovered that some of them were no longer alive," she says. Sylvia, who moved to England with her mother when she was five, has run the shop for the past nine years, and was responsible for making sure all the life stories were organised, rescuing them from piles of papers among her father's personal letters and shop admin. She arrived, aged 21, to help her ailing father out, and discovered that she was needed for more than just one summer. A sylph-like blonde, Sylvia has carried her own literary legacy around with her since birth. She was named after Sylvia Beach, the owner of the original Shakespeare & Company in the Twenties. An American in Paris, the first Sylvia Beach was friends with Ernest Hemingway, knew TS Eliot and published Ulysses when nobody had heard of a bedraggled Irish writer called James Joyce. Samuel Beckett was published in her in-house magazine, alongside Jean-Paul Sartre. Her bookshop was renowned, but after an internment of six months by the Nazis during the Second World War, Beach never reopened. So it was that George Whitman stepped in.

George, also an American living in Paris, arrived in the city to study at the Sorbonne, after a string of adventures. He'd travelled all over the world, studied at Harvard, been in the army, and run a little bookshop in Massachusetts, before setting off again, eventually settling on the Left Bank and opening Le Mistral, an English language bookstore. Later, he would change the name to Shakespeare & Company, in honour of the bookstore that was no more, and in homage to Beach's trailblazing spirit.

When Sylvia (Whitman) takes me up to see her father late one evening, he is in good form. Although it's after 11pm, he is wide awake, and in the mood to talk. Propped up in bed, his thin arms and chest protrude naked from beneath the sheets, a plate of food and a newspaper by his side. It is hard to reconcile the technology-fuelled existence on the Parisian streets outside his window with the world that made George Whitman. Many of the names he fraternised with have retreated into legend, but he is the thread that holds the link between that world and this. "Bill loved those books," he says, gesturing to a packed bookcase near his bed. Those are the medical books William Burroughs sought out over 50 years ago, and the ones Whitman still consults instead of going to the doctor. It's not a surprise, given his history, that George Whitman remains rebellious. He tells me that he doesn't like Sylvia's policy of allowing the Tumbleweeds to stay as long as they work two hours a day in the shop, as a way of both organising the store and letting them earn their keep. "It's exploitation," he says with his blue eyes sharp, fixing them on me. "People aren't supposed to work when they come here – they're supposed to see Paris, see the world!"

For all his complaints, though, he must be secretly pleased with how his daughter is running his literary bohemia. This summer, their festival, Festival and Co, was a tremendous success, a glittering affair with a string of intellectual heavyweights taking part – including Martin Amis, Will Self, Jeanette Winterson, David Hare and Philip Pullman – to great acclaim. There's a new literary prize for the novella, too, taking entries until December, and the day-to-day running of the shop always sees a steady stream of visiting authors who read their new works on the shop floor.

And despite George railing about Sylvia putting the Tumbleweeds to work, it's clear that they thrive on it. I later meet Chiara, aged 19, in George's living room (there's still a lot of bohemian coming and going here, which means that even at midnight you might bump into a Tumbleweed cooking dinner in George's narrow galley kitchen). "I love the smell up here," says Chiara, a Tumbleweed visiting for the second time from Italy. "It smells so exotic." She's right; it's a scent that can't be placed – maybe it's what comes from cramming so much tangible history and legend into one bijou living space. It's a popular place. If they let them, says Sylvia's assistant and bookkeeper Lauren, there would be a long queue of people wanting to make the journey up to the third floor to see George. Sylvia keeps a tight rein on the numbers, worried about tiring him. The actor Owen Wilson recently traipsed up the stone stairwell to walk through the mirrored living room and kitchen and into Whitman's book-lined bedroom. "I bring these people up to see George and introduce them as great actors, and dad just says, 'Oh. Have you written a book? No?' And then goes back to his paper," Sylvia laughs. "I think they like it though." Wilson is just the latest in a long line of people to have fallen in love with the bookshop. He was filming there with Woody Allen during the summer – a short outside scene which immediately attracted a crowd of around 500 people and a sea of mobile phones, all filming the process. (Whitman's dog Colette even managed to get into shot.) Ethan Hawke is a regular visitor – "I think Ethan is one of our real angels," Sylvia says. She's thinking of her father's founding philosophy, and the bedrock of the establishment. "Be not inhospitable to strangers, lest they be angels in disguise."

Freddie, a student from England, is in his late teens, and has already been here many times before. He looks a bit like an angel in disguise himself, with his blond unruly hair, flushed cheeks, and tweed jackets with elbow patches, but he tells me about his experiences with Jeanette Winterson, another "angel" in Sylvia's eyes, and a great supporter and friend of the shop. Winterson once gave Freddie 50 euros because she so enjoyed one of his writerly quips, and more money later when he was looking a bit hungry. Apparently she has done the same with many of the Tumbleweeds, offering the type of ad-hoc patronage that fits well with the spirit of Shakespeare & Co.

Because it is true that the spirit of the place is imbued with a certain kind of hope and generosity that is unusual in the normal run of things. It could hardly be otherwise: the walls are witness to thousands of individuals who arrived from all across the globe, took inspiration from the literature and the real-life characters they encountered in this crazy idyll, and left their life story on a page. It has been the setting for multitudes of people poised on the brink of adulthood, when the world was before them, and every possibility seemed within tangible reach. Tumbleweeds who stayed 40 years ago still remember it as a place where their dreams seemed more than just dreams – regardless of how their lives then unfolded.

The bricks and mortar cannot help but carry some imprint of all the souls that have passed through here. Shakespeare & Company holds the stories of thousands of lives, otherwise unmarked in literature, and George Whitman, centre of the vortex, keeps those stories alive.