“Inside Llewyn Davis” opens Friday at Point, Eastgate, Star Cinema and Sundance Cinemas in Madison. R, 1:45, four stars out of four.

If you see a man walking down the street in wintertime, freezing because he doesn’t have a warm coat on, you think one of two things: 1.) he can’t afford a warm coat, or 2.) he left it at home.

In “Inside Llewyn Davis,” filmmakers Joel and Ethan Coen show the reasons why their hapless folk singer hero, Llewyn Davis (Oscar Isaac) seems to be out in the cold in so many ways. Part of it is beyond his control — he’s a folk singer trying to make a living in early ’60s in New York’s Greenwich Village, a tough line of work for the great ones (and Davis doesn’t seen to be one of them). And part of it is a result of Davis’ penchant for self-sabotage, antagonizing allies and burning bridges and generally being a pill in the name of art. He doesn’t have a warm coat, but if he did, he’d figure out a way to lose it.

“Davis” may be the saddest comedy you ever see, or the funniest drama. It’s a note-perfect recreation of a time and a place, but builds that long-gone world just to tell one man’s story. And it’s a personal story of someone who is not very likable at all, refuses to let his guard down much, and yet we root for him. In other words, this is a Coen Brothers movie, odd and endearing, and one of their best.

The film follows a week in the life of Davis that’s a lot like a folk music “round,” in which various people drop in and out of the melody, singing different parts, and the song ends up right back where it started. Last verse, same as the first, a little bit louder, a little bit worse.

Davis knows how the song feels; he’s just barely scraping by, hopping from one friend’s couch to the other, living off the tip jar at the Gaslight Club and paltry royalties from his music. He watches, incredulous, as less talented contemporaries like the fresh-faced Jim and Jane (Justin Timberlake and Carey Mulligan) get a couple rungs higher up the ladder. His pained bewilderment is written all over his face, the expression of every promising also-ran in America. Why not me? Why them?

He road-trips to Chicago with a disdainful jazz musician (John Goodman, in the film’s one unabashedly hilarious performance) in the hopes of impressing a record producer. He comes back to New York. He considers quitting folk singing altogether and joining the Merchant Marines. Even quitting he can’t get right. Behind him he leaves a wake of hurt feelings and recriminations. We piece together that Davis used to have a singing partner, but he killed himself. How Davis feels about that, he never really says.

Isaac’s controlled, lived-in performance gives nothing away for free to the audience; Davis doesn’t ingratiate himself to the viewer any more than he panders to the Gaslight crowd. But we do get a couple of piercing glints “inside Llewyn Davis,” so to speak. One is how he treats Ulysses the Cat, a tabby belonging to some acquaintances who he loses, and then spends the rest of the film trying to find. Why does he care so much about a cat? Maybe Ulysses is meant to represent success, or happiness, or some other ephemeral thing we’re all chasing, just out of reach, a flash of orange in the crowd that keeps propelling us forward. Or maybe Davis just likes cats better than he likes people.

The other glimmer comes when Davis sings. Isaac sings about a half-dozen songs in the film, each to a different and very specific audience, and each time the performance reveals a little something different about him. The performances are beautiful, and rooted in his character.

I think the Coens, as much as they tweak Davis’ appetite for self-destruction, do kind of admire his dogged perseverance, his inability to quite give up on himself. The last line of the film is Davis sarcastically shouting “Au revoir!” which of course is a French way of saying goodbye. But the literal translation is “Till we meet again.”  Next verse, same as the first.