Anyone can own a gun in MA, BUT they cannot do so without obtaining either a Firearms Identification Card (FID) or a License to Carry (LTC).
Both are issued by the local police department and have to gun through the background checks, training and high fees. If you're turned down then no, you can't own a gun legally.
The catch to the whole thing is that the issuance of licenses is TOTALLY at the whim of individual police chiefs. If your police chief is anti-Second Amendment, you're screwed and there are plenty of them.
Obtaining a license here is an onerous, expensive, time-consuming and generally unreasonable process. But, once you get it, things kind of open up. At least we don't have stupid waiting periods and one gun per month limits like some states.
The only downside once you get your card is that the state AG has severly limited the types of news handguns we can buy..Glock for example