I live in Capitol Hill/Eastern Market and I love it. I'm actually leaving this summer to got to Minnesota and will really miss my neighborhood. Anywhere around Lincoln Park is generally going to be a great location. I used to live in Arlington (Clarendon) which was nice, but eh. Not especially unique or interesting to me. Here, I live around families, single people, young people, older people, people of all backgrounds and various income levels. It really is a nice little melting pot. There is always someone outside walking, running, or playing with their dog.
I am on Independence in SE and I would not go any farther over than 14th St SE. I know people who are fine living further over (closer to Stadium-Armory), but as a single female, I wouldn't walk over there alone at night.
Also, if you're around Capitol South, Eastern Market, or Potomac Avenue Metros, you are on the orange and blue lines. Believe me, as someone who lived only on the orange line before, this is a great thing at rush hour!