If the abuser has been convicted of a domestic violence misdemeanor or felony, can s/he keep or buy a gun?
No. Under Federal law, if your abuser has been convicted of a felony or a domestic violence misdemeanor, s/he cannot have or buy a gun.* If you're not sure if s/he has been convicted of a domestic violence misdemeanor, see What crimes are considered domestic violence misdemeanors?
In addition, MA state law says that if you have been convicted of a felony, a misdemeanor punishable by more than 2 years of jail time, a violent crime, or any weapons or drug-related offenses, you cannot have or buy a gun.**
* 18 USC 922 § (g)(9)
** Mass. Gen. Laws 140 § 131