What is it about Anna Karenina that gives it special status among the great novels? How is it that a sensational romantic tragedy of tsarist high society, interspersed with digressions into 19th-century Russian agricultural policy, written in a seemingly plain, straightforward style across 900 pages, still provokes both excitement and respect from readers as diverse as JM Coetzee, Jonathan Franzen and Oprah Winfrey, and lures Tom Stoppard to write the script for the latest of a dozen film adaptations? The book floats in some charmed section of the lake of literary opinion where the ripples from modernism and the ripples from Hollywood overlap without merging.

It is more admired than learned from. Anna Karenina couldn't be less like a conventional modern novel. Instead of a barrage of metaphors describing things in terms of other things that they resemble, Lev Tolstoy seeks the precise word for the thing itself. Instead of the solipsistic modern mode of events being experienced from the point of view of a single character, Tolstoy slips in and out of the consciousness of dozens of characters, major and minor. At one point he tells us what a character's dog is thinking.

Tolstoy doesn't believe in "show, don't tell". He likes to show and tell. The teller, the narrator of the book, is a formless, omniscient voice with no elaborate Rothian construct to justify his role. No first-person or free-indirect speech here. Even while we're in a character's head, it's the narrator who recounts the character's experiences through liberal use of such unfashionable phrases as "she thought", "he felt" and "it seemed to him that".

Tolstoy creates a space for the narrator's independence – the narrator is close enough to the characters to rely on them for his existence, but free enough to pass unchallenged judgment on their actions, and to tell us things about them that they don't know about themselves. The most powerful passages are those where Tolstoy slows time down to note each thought, gesture and feeling of Anna and her lover Vronsky, with a third entity present – the narrator – not only lodged deep in the two psyches, but standing back to tell us the ways in which one is misunderstanding the other.

Each time I reread Anna Karenina, picking my way past the attics and cellars and rusting machinery of Tolstoy's obsessions and prejudices, a new layer of his craft emerges, to the point where, for all my admiration of Joyce, Beckett and Kelman, I begin to question whether the novel form isn't too artisanal a medium for the surface experimentation of the modernist project ever to transcend the flexing of space and time that apparently conventional language can achieve in the hands of a master.

I'd noticed before that Tolstoy, whose characters spend so much time in Moscow and St Petersburg, barely describes these cities. Reading Anna Karenina again, I see that it's more extreme than that; urban buildings and landscapes are practically invisible, whereas the countryside is described in exquisite detail.

To Tolstoy the city is a static, artificial place. It is as if he does not believe cities are permanent, as though he feels that if he ignores them, they'll go away. It turns out that everything Tolstoy cares about, everything he describes taking place outside the character's heads, is alive and moving, in the non-human world of dogs and horses and leaves as in the human world. No human action is too small to be recorded: Karenin's knuckle-cracking, Anna screwing up her eyes, Vronsky touching the ends of his moustache. The characters are always smiling, frowning, blushing, twitching, fidgeting, touching, kissing, bowing, sobbing, and deconstructing these signs in each other. They come to us alive with intentionality, describing themselves in movement, waltzing through the ballroom, trudging through the marsh after wildfowl, racing horses, cutting hay.

As busily as Tolstoy's creations move through space, so plausibly they move through time. How hard it is in narrative fiction, be it novel or film, to represent the chaotic reality of the passage of time, when the way a person acts or thinks one moment doesn't necessarily have a direct connection to the way that person acts or thinks 10 minutes later, or the next day, or for the rest of their life. No other novelist I can think of takes the risks Tolstoy does with the readers' understanding of what his characters are by allowing the characters to be so true to the emotions of each particular moment, even when those emotions contradict the overall portrait. The most odious characters are never beyond momentary redemption, and the most admirable characters must endure patches of vileness.

One harsh, simplistic, but not entirely inaccurate reading of Anna Karenina is as Tolstoy's justification of his life up to the moment when he wrote it, through the character of his alter ego, the chippy, idealistic landowner Levin (Levin = little Lev), whose journey to faith, family and contentment down on the farm acts as a counterpoint to Anna's path of extramarital passion and death in the Babylon of the urban beau monde. Yet Tolstoy doesn't spare Levin, the character with whom he is most in sympathy.

When Levin is out shooting with a friend at dusk and summons the courage to ask after Kitty, the young woman he loves but who turned down his offer of marriage, he learns that she's still free and is seriously ill. At this moment of high drama and revelation, two woodcocks fly over, and he forgets about Kitty in the excitement of shooting the birds. "Now, what was that unpleasant thing?" he thinks afterwards. "Oh yes, Kitty's ill." Back at the house he admits to himself that while he's glad she's still available, he's even more pleased that she's sick; serves her right, he thinks.

It's not attractive for Levin to feel this schadenfreude towards the woman he wants to share his life with, or to have the overflight of a small game bird blot out all thoughts of her just when he's heard she might be dying. But Tolstoy has the confidence to relay these secret moments of unlove, certain – rightly – that by being true to his weakness in one particular instant in time he will make Levin more real and human without poisoning the instants of time to come, when Levin will show himself more like the man he wants to be.

All Tolstoy's mastery of time, space and language come together in a single moment in the middle of the book, when Anna's estranged husband Alexei Karenin, a dry, stiff government minister, and her lover Vronsky, a handsome young cavalry officer, meet beside the bed where Anna lies gravely ill after giving birth to Vronsky's child. Grief-stricken and ashamed, Vronsky is covering his face with his hands; Anna orders her husband, who is also weeping, to pull the hands away and expose her lover's face. With that gesture, Anna effects a reversal in the status of the two men. Vronsky, who had despised Karenin because he wouldn't fight a duel, is now humiliated and dishonoured; Karenin, flooded with forgiveness for everyone, wins back Anna's respect. In that moment of time, with Anna seemingly dying, the transformation is quite real. But time shifts, and the old reality comes back. Anna gets better and hates Karenin more than ever for his forgiveness. Vronsky restores his honour by shooting himself (he misses). The arc of Anna's destruction resumes. In the novel there are no turning points, only points, and characters travelling through them.

For a spacious novel so concerned with families there's a mysterious absence at the heart of Anna Karenina. The heroine has no childhood. She comes equipped with a son, a dull older husband, a brother, friends, a place in high society, but no past, no younger self. There is no description of how she came to be married. Her parents are, presumably, dead, and are never mentioned. She is fully formed, ready to fall in love with the dashing Vronsky.

It's not just Anna. Most of the other principal characters have no forebears on the scene. Levin was, like Tolstoy, orphaned at an early age. Vronsky's mother is occasionally present but when we first encounter him Tolstoy quickly tells us: "Vronsky never knew family life."

Although children as characters are present only in the background (with one brief exception), the book is preoccupied with the parent-child relationship: with having or not having children, with choosing between paternal-maternal and romantic-sexual love, or working out what to tell children when they ask what life is for. And the novel is about children in a deeper way, one that speaks to the stretched-out generations of the rich world now, where people in their 20s, 30s and 40s expect to have parents who are still alive and constantly reassure each other that they are young – that they are, in effect, still children.

Anna, Vronsky and Levin are in their early 30s, young in today's terms, but Tolstoy doesn't provide them with an earlier generation to backstop them, or to be remembered. They are obliged to stand independently as grown men and women. This means following an existing set of social rules, like Vronsky ("One must pay one's gambling debts, but need not pay one's tailor; one must not tell a man a lie, but one may lie to a woman"), or breaking rules, as Anna does, or inventing their own set of rules, as Levin tries to do. They can have children – they should, in Tolstoy's view, have children – but they cannot be children. However, among the principal characters, there is an intriguing exception: Stiva Oblonsky.

It's the Oblonskys, not the Karenins, who are referred to in the novel's famous first line: "All happy families are alike, each unhappy family is unhappy in its own way." And the Oblonskys, Dolly and Stiva, are unhappy because Stiva is screwing around. Like the other main characters in the book, like Tolstoy himself, the Oblonskys are aristocrats, with the trappings of the upper class – rank, servants, a town house and a place in the country. But they're in debt, and the country house is falling to pieces.

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Where Dolly, a kind, pious, modest, anxious figure, the mother of five living and two dead children, belongs very much to the old Russia, Stiva Oblonsky, her husband, is recognisable as the caricature of a modern man. Stiva is an old Russian short form of the name Stepan, but I can't help thinking of him as a tanned guy called Steve in a pink open-necked shirt. Beloved by everyone for his charm, his healthy glow and his radiant smile, he's generous, gregarious, greedy, hedonistic, trivial, shallow, fond of gadgets and sex with non-threatening women, infantalised by fashion and marketing. He serves six different kinds of flavoured vodka at his parties. He reads a liberal paper not because he's a liberal but because it suits his lifestyle. He uses his connections to get a cushy, well-paid government job and is selling off his wife's properties cheaply, yet still struggles to afford the life he thinks he deserves. "However much [Stiva] tried to be a caring father and husband," the narrator tells us, "he could never remember that he had a wife and children."

The clashes between the moralistic Levin and his friend Oblonsky, sometimes affectionate, sometimes angry, and Levin's linkage of modernity to Oblonsky's attitudes – that social mores are to be worked around and subordinated to pleasure, that families are base camps for off-base nooky – undermine one possible reading of Anna Karenina, in which Anna is a martyr in the struggle for the modern sexual freedoms that we take for granted, taken down by the hypocritical conservative elite to which she, her lover and her husband belong.

That elite does exert a growing influence as the book unfolds, and it is true that the moralistic side of the establishment prevents Karenin showing Anna mercy. A case could be made that the unhappy family of the opening is the Russian aristocracy in the 1870s, trying to hold the line against excessive change after the grant of freedom to millions of human beings it had owned as slaves, the peasant serfs, in 1861. The principal characters in Anna Karenina are literally part of one big formerly slave-owning family. Levin marries Kitty, who's the sister of Dolly, who's the wife of Stiva, who's the brother of Anna, who's married to Karenin. Even Anna and Vronsky are distantly related; their cousins are married to each other.

The tragic consequences of the pursuit of love for love's sake, in defiance of the rules laid down by one's peers and one's family, is an eternal story, and that story is in Anna Karenina, but that story is not, by itself, the book Tolstoy wrote. Anna Karenina is no Romeo and Juliet story of star-crossed teenagers unjustly destroyed by their elders' cruel laws, but a story of adults vexed by boundaries. It is the portrayal of a clash between an old world of rigid religious codes, duels, fixed gender roles and strict class division and a new world of divorce, separation, custody battles, women's self-determination and uncertain moral rules.

It's not that Tolstoy sympathises with high society's mixture of moral outrage and gladiatorial blood lust over Anna and Vronsky's affair. While it's true he allows Anna not a moment of sexual pleasure, he had censors to contend with, and makes it clear how unsuitable a partner for Anna her husband is. As the book goes on, in step with Tolstoy's increasing religiosity and his disenchantment with the project, he does put an increasing and sometimes oppressive emphasis on women's role as mothers. But none of this means he ever loses compassion for or patience with the painful, intricate detail of Anna's dilemmas.

Anna's love for Vronsky is a nobler affair than the infantile sexual consumerism embodied in Stiva Oblonsky, the emblem of modernity. Yet for Tolstoy the line between sexual freedom and sexual greed is not a clear one. He looks ahead to the era we live in now, where the dragon of sexual repression has been slain and sexual freedom prevails, and where, better as life is, we haven't rid ourselves of the reasons Anna throws herself under a train. A woman may still marry a man she doesn't love, still feel shame and guilt for having an affair with someone else, still hate him for forgiving her, still (more rarely, certainly) lose custody of her son, still find that people she thought were her friends side with the husband, and still find that the man for whom she left the husband, the man she loves sincerely and passionately, doesn't understand her at all.

I'm not sure Tolstoy ever worked out how he actually felt about love and desire, or how he should feel about it. He was torn between compassion and moral rigour, between lust and self-denial, between loving his wife and being bored by her. His uncertainty is reflected in the dual portrayal of his wife in Anna Karenina – as the virtuous, somewhat frumpy Dolly, worn out by childbearing, like the woman his wife was when he was writing the book, and as the feisty, pretty teenager Kitty, like the woman his wife was when he married her. They must have seemed to contradict each other, yet each was true to her time; and Tolstoy, for all that he was a master of time, was only a slave to truth.

• James Meek's novel The Heart Broke In is published by Canongate on 30 August. Joe Wright's film of Anna Karenina opens in the UK on 7 September.