Is it illegal to not carry an ID in Massachusetts?

Your question is vague. Not carrying an ID is not in and of itself illegal.

You need an ID if you need to prove who you are or prove your age or board a plane. If you commit a crime or encounter the police they will want to see your ID in order to establish your identity. If you have your taxes prepared the tax preparer will need to see your ID. A bank will require your ID to cash your checks or allow withdrawals. You need a valid driver's license if you drive a motor vehicle.

Otherwise you do not need to carry an ID.