New York City
The setting takes place "not long ago" in "uptown New York," which gives us a pretty good starting point for parsing the specifics. After all, "The Magic Barrel" keeps things spare, so we have to infer certain things. The story was published in 1958, so we're probably pretty safe in assuming that it takes place in the early 1950s.
That's awesome, but what does it mean? Well for starters, the setting gives it a tone and a rhythm that you wouldn't see in a story set elsewhere. . There's a general claustrophobia, as you might expect from The Big Apple. Leo livers "in a small, almost meager room, though crowded with books" (1), which is part of a "dark fourth-floor hallway of the graystone rooming house" (2). Not a lot of sunny spaces in Leo's life.
Salzman, similarly, lives in "a very old tenement house" where "Leo found Salzman's name in pencil on a soiled tag under the bell and climbed three dark flights to his apartment" (145).
Right away, you can see those New York-y clichés piling up. Tight spaces. Tenement houses and apartment buildings where you're never alone. There are a lot of people and a lot of pressure, which helps contribute to Leo's overall stress level. You can't get that kind of stress in, say, Oahu.
But at the same time, we also get a sprinkling of magic amid the crowded buildings. When Finkle first talks to Salzman, he
"… observed the round white moon, moving high in the sky through a cloud menagerie, and watched with half-open mouth as it penetrated a huge hen, and dropped out of her like an egg laying itself." (4)
Later, when he goes to meet Stella, "Violins and lit candles revolved in the sky" (201). There is a little fairy dust here. It's not quite magical realism, but it gives you the sense that something special is going on here: something that makes all the fishy smells and tiny apartments and general New York cruddiness worthwhile.
It's also worth noting that these magic images mostly seem to appear in the sky. They're a part of New York, since they affect the seething masses of New York residents, but they're also detached and a little distant. They speak to a freedom and an expansiveness that Leo presumably longs for. Also, the fact that they appear in the sky suggests some kind of heavenly origin: that the magic comes from God and that God's actions influence the story.
Finally, the specifics of this New York neighborhood—uptown Manhattan, which has a lot of Jewish communities—is reflected here and there throughout the text. The early references to the Forward (1), a Jewish newspaper and Yeshivah University, (1) a school with a history of rabbinical study, underline the cultural realities of the story.