How long do you have to change your mind once a contract has been signed for a new car in Massachusetts?

Right up to the moment the contract is signed.

A lot of people believe that you have some time to "change your mind" when you buy a new car (some of them believe this adamantly and rather stridently); however, this is not, in general, true in any state in the US. (Most states do have some kind of "cooling-off period" required on certain types of sales such as those made by door-to-door salespeople, but car sales are typically excluded from this either explicitly or because nobody goes door-to-door trying to sell cars.) Some states do require that buyers be offered this as an optional add-on for new car purchase contracts, but the dealer is allowed to charge extra for this, and if your contract doesn't explicitly say that you have X amount of time to change your mind, then you don't.

Note that even though the state does not force a dealer to allow you to back out of a contract, the dealer can choose to allow you to do so anyway. You can always ask.