I'm just finishing a masters program in Boston and have been living here for almost three years. It is expensive, but there are ways for it not to be so bad. I live in Somerville and pay $655 a month for one quarter of a 4-bedroom apt shared with 3 roommates (total price for the apartment is $2570/month), and that price is considered a good deal here. Living with so many people isn't ideal but we're all grad students and really busy, so we don't get in each other's way much! Which I think is a pretty common situation here.
My advice for living cheaply in Boston would be:
Live far from the T: usually the closer you are to a subway stop, the higher your rent will be. I live a mile from the nearest red line stop and my rent is a few hundred dollars cheaper than in the area closer to the T. I don't mind the distance--it's what I have instead of a gym membership . In total my commute downtown is 15 mins walking and 15 mins on the red line, not bad at all.
Live with roommates: if you want really cheap, I don't really think it's possible to avoid roommates. I've never heard of a studio going for less than $1000 even in a bad area ($1200 in a good area) or a 1br for less than $1500.
Take the tiny room: I've found a lot of 3-4br apartments here, especially in Somerville and Cambridge, have one bedroom that's really small, and many people will be willing to let the roommate who takes that room pay less (my apt has one tiny bedroom and the girl who lives in it only pays $505/month, which is fine with the rest of us who had too much stuff to put in the tiny room).
Live in Somerville instead of Cambridge, Brighton instead of Brookline... But avoid Allston if possible (very many drunk undergrads, all the time). I actually really love Somerville. It's a more reasonable version of Cambridge that has a lot of fun stuff to do. The area I'm in is called Inman Square, on the Cambridge/Somerville town line, halfway between Harvard and Central Squares. Inman is a fun and cute but cheaper neighborhood than the other Cambridge/Somerville "squares" (Harvard, Porter, Central, Kendall, Davis). Other cheaper areas in Somerville far-ish from the T are Union Square, Magoun Square, and Teele Square.
I have heard of people living cheaply in Dorchester or Roxbury but have never felt particularly safe being there... Anyone feel free to correct me on that though.
Even with the expense and the inconveniences you have to deal with to live cheaply in Boston, I love this city and have really enjoyed living here as a grad student. Feel free to ask me anything else about it!