$2k will get you quite a bit. You could get something in the South End which will be pretty hip and happening though you should definitely try to be gay, if possible. It'll make things easier.
The North End likewise, but if you have a car it'll suck and there's a pretty serious stigma against "outsiders" looking to get in on the scene.
Central Square Cambridge matches your criteria.
timmhaan, yeah it's hard to believe because Somerville is nearly entirely Boston 3-deckers, but there's very little open space and it's overwhelmingly residential (which makes the city's finances a real challenge). Around the turn of the 20th century it actually boasted a higher population density than Calcutta.
Like Rome, Somerville was founded on 7 hills, the tallest of these being Prospect Hill. Following the Battle of Bunker Hill, the rebel army retreated to Prospect and erected what became the strongest fortifications in the battle for Boston. In 1776, the Citadel saw the flying of the first American flag.
It's a real ***** on a fixed gear.