I dont know if this matters much, but I just spent past 3 months doing various rotations in this city. Let me start off by saying that this city is expensive, especially for me who is used to Midwestern prices and cost of living.
That being said, if you really look for a place, you can find an apartment in periphery of Boston (Malden, Brighton, Allston, etc) with 700-900 a month, with roommates of course. Also, having a car will NOT work in this city. All the places that I was a sub-I required met to pay a daily fee to park which obviously adds up. Take the train and the buses. I used to wake up at 4:30 to catch the bus so that I can reach at 6am. That saved a lot..
In a nutshell, there are ways you can save money and find good places, but you have to find those!