It depends on what you consider commutable distance, I suppose...
I live in Astoria, Queens, at the Broadway stop on the N/W line. It takes about a half an hour to get to Cardozo directly (including walking). If I transfer to the 4/5 at 59th/Lex, it cuts off between 5 and 10 minutes. In this area, you could probably find a garden level studio (basically basement, with window at ground level) for around that price. If you go into less gentrified areas of queens, you'd definitely be able to do it, though the commute would be longer. I'm sure the same is true for Brooklyn, though I don't know areas outside of Park Slope and the Heights well. The nice thing about Union Square is that there are a lot of trains stopping there so it's easy access.