That's a tricky thing for us moderns, who tend to define "true" things as "events that actually happened in our universe in space and time." But another kind of "truth," driven by narratives we tell one another and take in together, can also shape reality in powerful ways. Consider how American stories about cowboys have shaped the way Americans think about themselves on the world stage: self-sufficient independent individuals wearing white hats and riding forth to serve justice.
The Coens have been drawing on legends and archetypes (especially those familiar to Americans) since they started making movies. Blood Simple (1984) traces its lineage through noir classics like Double Indemnity (1944). Raising Arizona (1987) features a hell-raising motorcyclist who seems to be the kind of demon of indeterminate origin that Javier Bardem would later portray in No Country for Old Men (2007). The Big Lebowski (1998) is enough of a head-scratcher that it's been interpreted as everything from political commentary about the Gulf War to a fable about a latter-day messiah. True Grit (2010) is a remake of a classic Western, which originally starred the most iconic of all cowboys, John Wayne. And A Serious Man (2009) is transparently the Coens' remake of the story of Job, with a twist ending.
One of my favorites, Fargo (1996), both capitalizes on the modern penchant for "true" stories and inverts it. The film's title cards claim that "THIS IS A TRUE STORY." But the brothers admitted later that the whole plot is fictional, though based on various crimes drawn from the newspapers. "We weren't interested in that kind of fidelity," Joel said in an interview. "The basic events are the same as in the real case, but the characterizations are fully imagined . . . If an audience believes that something's based on a real event, it gives you permission to do things they might otherwise not accept."
Inside Llewyn Davis bears a cosmetic resemblance to the most obviously mythical of the Coens' films, O Brother, Where Art Thou (2000), with its dual backdrop of an Americana soundtrack and references to Homer's Odyssey. Both films fit the larger narrative type to which Homer's poem belongs: the hero goes on an epic journey, then returns home (at one point Llewyn even spots the movie poster for The Incredible Journey, though it's a bit of an anachronism, since it actually came out in 1963).