At night, the ancient theatre at Epidaurus has the quality of an etching of the human brain, a tiered slice that fills the eye as one stands on the stage and is stunned by the silence and the smell of cedar. I find myself feeling that, on top of this mountain two hours from Athens, the slide rule of time does not apply. One is entirely in the past and in the present at the same time.

During the famous drama festival that takes place here every year, Greek classics are performed to breathtakingly large audiences. There were 12,000 people at a performance of Electra I attended, sitting on the stone steps of this amphitheatre on the site of the most celebrated healing centre of the ancient world.

This year, the festival has broken with tradition and featured not only the classics, but also modern plays that were inspired by, or relate in some way to, Greek drama. We were asked to bring over the National Theatre's production of Samuel Beckett's Happy Days in which I have been playing Winnie, a woman who sits buried in a mound of earth in bright, never-fading sunlight. This has caused a scandal in the press, with conservative elements asking why a modern play, particularly a Beckett play, should be staged in this ancient monument. What can it have to do with the Greek tradition?

At a press conference to give the performance a push, this is clearly an issue. As we respond to questions, a helicopter flies over carrying a giant sack of water to quell yet another forest fire, and I think of Winnie's line in the play: "With the sun blazing so much fiercer down and hourly fiercer is it not natural things should go on fire never known to do so."

Travelling down the mountain, past the dry olive groves and the parched earth that is Greece this summer, I hear her again: "Might I myself not melt in the end perhaps or burn, I do not mean burst into flames just bit by bit be charred to a black cinder, all this visible flesh."

In Greek drama, characters are bound by family, location and history, as well as a shared understanding of a moral code and a sense of common purpose. Beckett offers a much bleaker world with no such ties. Winnie is as far away from those Greek icons as one can imagine, and yet there are connections between the two very different traditions, not least to do with language and silence.

Winnie is static in earth, and talks to "break the silence of this place". Even though Willie, her husband, who lives in a cave behind her mound, rarely replies, she remains optimistic about their marriage: "Just to know in theory you can hear me even though in fact you don't is all I need."

Winnie is frightened of losing Willie and being alone: "I am not merely talking to myself that is in the wilderness a thing I could never bear to do for any length of time." She is afraid of silence, not the silence I experience here in Epidaurus, which is an unfathomable silence that might reform you - a silence worth taking back to the city - but one more desperate. "Not another word as long as I drew breath, nothing to break the silence of this place." Happy Days deals not with the grandeur of tragic issues, but with the minute detail of our anonymous suffering. We have banished all gods, Greek and monotheistic, and in Beckett's play we are stuck in the ruins of our planet, gazing at the pitiless sun, which is "the bell for waking and the bell for sleep".

The actress playing Clytemnestra at the festival told me she approved of the introduction of 20th-century plays. It will be an inspiration, she said, the beginning of a conversation between the classical and the modern: they have run out of ideas for the classics. Notwithstanding the dreadful subject matter of Greek tragedies, the language in these plays is confidently assumed to be able to forge morality. The language of Beckett is quite the opposite: it is assumed to have failed us as individuals and as a species, and morality is whatever is left over.

Will Happy Days say to the Greeks we are at the end of something? Does the play prompt us to look forwards or backwards? As I look out from the stage at an audience not smaller than the population of a small town, I will think of Winnie's "strange feeling as if someone is looking at me".

I sit at midnight with Tim Potter, who plays Willie, as lumps of our modern set are brought into this ancient space. We marvel that this gem of Beckett's, a play that discusses and celebrates the "stuckness" of the human form, should be having a showing in this remarkable theatre to a larger audience than it has ever received. Yet the play and, indeed, we cast members feel really comfortable here. This is a place of total serenity, built so beautifully, and we think it will serve to help us communicate.

I often feel that, in the world of Big Brother and the barbarism of many of our cultural choices, theatre has lost its way. But Beckett works against the trend and rescues theatre. The Greeks took us beyond language to pure emotion; Beckett takes us beyond language to silence, the silence we all will and must face. We may find it holds desolation, but here one feels that, against all odds, it holds promise.