Hello Jill and Todd- I'm a Mass. licensed builder, and also a RE investor. I've been through this question many times.
Other than officially callling it an apartment, if the town doesn't allow it - via the zoning ordinance- there may be other options, like an in-law apartment or an "au-pair" suite.
Typical Mass. state building code is that the finished ceiling height must be at least 7'0, for a finished basement area. If you don't have that (after sheetrock and a finished floor), I'd give up. There are other building requirements, and it would be smart, and free, for you to have a local general contractor do a walk-thru of your house and basement and give you a written quote of what you want to do. He will know if its feasible after a walk thru, etc. And then you can see roughly how much it will cost anyways to do.
That's my 2 cents. If you need anything else, please feel free to contact me directly.