Of course, Walter's fantasies take us elsewhere, but we'll get to that in a minute. Waterbury is a pretty big city in Connecticut. Though Thurber never mentions the state, just the city, we can take a pretty solid guess that he's talking about the only major Waterbury close to the Tri-state area. Also, a newsboy goes by shouting about the Waterbury Trial, which pretty definitively refers to the Waterbury Trial of 1938, which took place in CT.
OK, enough geography. The real-life setting of this story is pretty mundane: a hairdresser, a parking lot, a hotel lobby, a drugstore – all everyday elements of any town or city. The banality or dullness of these locations reflects the dullness of Walter's everyday life. This is greatly contrasted with the settings of Walter's fantasies: a "Navy hydroplane" in a storm, an operating room, a courtroom, a dugout, a wall before a firing squad. These settings are dramatic, exciting, and out of the ordinary.