It sounds as if you plan to move without having a job lined up ahead of time, just hoping that you can find something. Many people would advise you not to do this. However, if you are going to make such a move, you at least want to try to reduce the risk.
One question I have is why you are thinking of moving to Boston. The Boston area has a lower cost of living than NYC, but Boston is still one of the more expensive cities in the U.S. Cost of living depends some on the neighborhood or town where you live, and it's true that generally it costs less to live in a suburb well away from the city--like Framingham--than to live in or very near the city. Still, anywhere in the Boston metro area is expensive compared to most of the U.S. If you were unable to find a job quickly, any money you have saved which you could live on for a time would not last as long in Boston as it would in most cities in the South or Midwest, or many interior regions of the West.
The next question is why you are considering Framingham in particular within the Boston area. Framingham has one obvious advantage for someone who is planning to move and then find whatever job might be available. That advantage is that Framingham's local economy is dominated by malls. There are lots of restaurants and large stores that are potential places of employment.
Framingham also has disadvantages for someone in your situation. Possibly the most obvious problem for someone without a car is that Framingham covers a large area. There is a local bus system that covers Framingham and several nearby towns, but these busses do not run in the evening, so anywhere around Framingham you might want to get by bus you would need to travel to during the daytime. I'm not sure, but I think it may be that these busses do not run at all on Sunday, maybe not on major holidays either. So it's nice to have some local public transit, but it's still not something you want to have to rely on for all your transportation.
It IS possible to find rental property in neighborhoods near the malls, making it possible to perhaps find a job in a mall, and to do your basic shopping there, all within walking distance of where you live. That is possible, but there is a lot of Framingham nowhere near the malls, so you would be limiting your options on where to live if you had to live near the malls. The malls also are not really close to commuter rail, which would be the best means of getting to places outside of Framingham. You could walk from the areas around the malls to the train station if you really had to, but it would be a very long walk, most likely at least an hour. So, if absolutely necessary, it would be possible to live in Framingham without a car, but it's not the best town to live in if you plan to walk everywhere or take public transit.
I also have some concern about your plan to find a rented room. I've lived in Framingham, in rented rooms, and had some bad experiences with small-time local landlords in Framingham. I've known other people who have had bad experiences there as well. Here is a link to a past thread where I went into some detail about this (my post is post #2):
http://www.city-data.com/forum/massa...am-radius.html.
Unfortunately, this kind of situation may occur with a lot of small landlords everywhere. However, I know from bitter experience that it is a problem in Framingham.
There are some other towns in the Boston area that I might recommend. Which town might work would depend on your budget, and the details of your situation, such as how much walking you are willing to do. You might check out Natick, right next door to Framingham, also a town with a lot of mall shopping, a nicer town than Framingham overall, and a town where it's at least possible that you could find a place to live that would be at least sort of close to both the malls and the train station. Depending on your budget for housing, you might look at Newton, an expensive town, but maybe one where you could find a small rental property like a room for a somewhat affordable price. Newton's businesses are mostly small shops in local small commercial districts, so there are not as many potential employment options as there are in Framingham or Natick, but Newton is very convenient to Boston by public transit, so the employment options in Boston would be very close by. There are other towns that might work as well, depending on the details of what you are looking for. Natick and Newton are two towns I'm familiar with which happen to immediately come to mind as possibilities, again depending on your particular needs in a place to live.
I wish I could be more encouraging. If absolutely necessary, it would be possible to live and work without a car in Framingham, but it would not be easy, especially not unless you would be prepared to do a lot of walking. You also need to really consider your reasons for wanting to move anywhere in the Boston area, because the area has a high cost of living, so any money you have saved would not last as long as it would in some other cities if you did not find work right away. There are possibilities for someone in your situation in the Boston metro area, but you need to give some serious consideration to how you plan to make this work before you move with no job and no wheels to a town that is not really so good for getting around on foot in one of the more expensive metropolitan areas in the U.S.