In Massachusetts, you can keep a non-large capacity handgun in your home or place of business if you have a valid Firearms Identification Card. Pursuant to G.L. c. 140 § 129B, you are prohibited from being issued a Firearms Identification Card if you have been convicted of a misdemeanor punishable by imprisonment for more than two years. However, there is a restoration of civil rights provision, pursuant to which your right or ability to possess a non-large capacity rifle or shotgun can be restored 5 years after conviction or release from incarceration or supervision, whichever comes later.
If the crime of driving under the influence carries a sentence of less than 2 years, then you can apply for a Firearms Identification Card, which would allow you to keep a non-large capacity handgun, rifle, or shotgun in your home. You could not carry these weapons and you could not purchase a handgun without a permit to purchase.