In the "Setting" section, we explain a bit about the climate of 1797. The story takes place during the Napoleonic Wars in the same year as a number of massive mutinies in the English fleet. Thus a man like Captain Vere was concerned not only with fighting a war against the French but also with keeping peace within his own crew.
As much as the narrator tends to zoom out to explain the historical circumstances of the story, he tries to confine himself to the events on board the H.M.S. Bellipotent. As he says, the story is restricted "to the inner life of one particular ship and the career of an individual sailor" (3.1). At another point, the narrator notes how the ship itself makes up the story's entire stage. He says, "In the present instance the stage is a scrubbed gun deck, and one of the external provocations a man-of-war's man's spilled soup" (13.1). The confined space of the Bellipotent is where the story happens, where the events take place.
In making this point, the narrator is concerned with the fact that even the most dramatic of passions can unfold on a minor stage. As he says, "Passion, and passion in its profoundest, is not a thing demanding a palatial stage" (13.1). The Bellipotent, as you've probably noted, is anything but a palatial stage. In fact, the nature of the Bellipotent as a stage is extremely important to the story, to how events unfold. In other words, the Bellipotent is not just a passive setting where things take place, the setting actually affects the way that things happen.
How does the ship do this? One way is that it is an extremely confined space. Claggart, for whatever his reasons, has a consuming hatred of Billy Budd. In normal circumstances, he might just go out for a drive in the country to get some fresh air. In this case, though, he has nowhere to go. He is constantly forced to rub up against a man that he can't stand. Whereas in most plays, it is implied that the action is taking place in a location much wider than the stage itself, here the "stage" sets the boundaries of where the characters can actually move. The fact that Claggart cannot physically escape from Billy means that he cannot escape from his own hatred, and the confined space is one reason that Claggart snaps and falsely accuses Billy.
The other thing about ships in general is that they are isolated things, that they're floating out at sea, and the Captain is using his little compass and his knowledge of the movements of the sun and the moon and the stars to stay on course. Things are not solid out at sea. There's a third axis that doesn't exist on land. Things go down into the water to unfathomable depths. In the case of the Bellipotent, the narrator is careful to note that when the events unfold the ship is out on a separate mission and hence is alienated from the rest of the English fleet. The sense of isolation contributes to the enormous responsibility and helpless that Captain Vere feels when he is confronted with the case of Billy Budd.
Claggart, clever devil that he is, knows this and takes advantage of it. He doesn't approach Vere until they are out on their own, and he waits until they've just failed in their pursuit of a French ship. He waits until Claggart is "doubtless somewhat chafed at the failure of the pursuit" (18.3), and then he takes advantage of the setting. He waits to accuse Billy until Claggart is especially vulnerable and alone.