Let's start with the time: even though Fitzgerald doesn't come out and say what year it is, his story is very much written in the here and now. Based on its original date of publication (May, 1920), we can guess that readers were supposed to believe that it was happening right then.
What's significant about this time? Hah! An easier question might be "What's not significant about this time?" First of all, the Roaring Twenties were a famously tempestuous period in American history. Despite the fact that the Temperance (anti-alcohol) movement was in full force – it succeeded in 1920 with the beginning of Prohibition, which lasted until 1933 – a new generation of young people was making waves in society with their wild parties and radical ideas about social norms. One of the most exciting and provocative changes was the development of a new kind of woman, one who talked and partied like the guys, wore scandalously short skirts, and refused to be bound up by traditional female limitations. This is a conflict we see emerge strongly in "Bernice," which reflects the greater trend Fitzgerald witnessed in his social circles.
Now, on to place. This story unfolds in some unnamed Midwestern town or city. We can locate it in the Midwest because of a few reasons: first of all, Bernice refers to her hometown, Eau Claire, many times. Eau Claire, Wisconsin is the only Eau Claire we can think of; Bernice never feels the need to clarify what state she's from, so that's our guess. For this very same reason, we think she's not too far from home – she never has to explain where she comes from to people. Secondly, our location is clearly not on the East Coast, since everyone makes a big fuss about going out East to college. Finally, since Fitzgerald himself was a Midwestern boy (he came from St. Paul, Minnesota) and often places his stories and characters in and around his homeland, we think the Midwest is the most likely location. This makes some of the thematic elements of the story make a little more sense – even though the flapper might have been a common sight in New York, Bernice's bobbed hair makes a big, huge splash in this city, which might be a little behind the times.