Since your top priority is proximity to MGH then the nearest neighborhood is the West End. It's 2-5 minutes walking distance to MGH and gives you a tucked away community feel while offering all the conveniences of city living. Second would be Beacon Hill.
Lucky for you Boston is one of the greatest walking cities, a fabulous public transportation system with a combination of trollies and busses. Get a T map and start exploring. When you are ready to focus on a particular neighborhood check out my website at http://www.mikefeldman.com. Call the mover and enjoy all that Boston has to offer.