Take a story's temperature by studying its tone. Is it hopeful? Cynical? Snarky? Playful?
Even though in a play tone isn't as obvious as it might be in a short story or novel, we can still kind of see, by the way the whole thing is structured, what the author's attitude is toward the sad lives of the characters. And we think that the anticlimax is the perfect way to see that Chekhov is totally making fun of Vanya and the gang. Look at the drama, and tell us this ain't comedy gold:
VOYNITSKY: Let me go, Hélène! Let me go! [Freeing himself, he runs in and looks for Serebryakov.] Where is he? There he is! [Shoots at him.] Bang!
Haven't I hit him? Missed again? [Angrily] The devil, devil… devil take you. [Hurls the revolver on the floor and sits down on a chair exhausted.(. . .)] (3.430-36)
Try to picture this scene in your head. Here's some Uncle Johnny dude running around and actually shouting "Bang!" while he's trying to shoot someone, only to miss twice and drop into a chair exhausted like a silent-movie actor on a bad day, having achieved nothing. You know what that is?