Overview
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Eastern Market is a neighborhood centered around (and titled after) Eastern Market in Washington DC. It is located a few blocks from the Capitol. It is a somewhat bohemian, and thanks to local residents has survived the modernization that the rest of DC has been experiencing. It is located in Southeast, an area I would generally recommend steering clear of except around the Market. It's a friendly place with alot of history.
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Pros:
Cons:
Recommend? 
As I said, the people tend to be a little more bohemian than you might find elsewhere in DC. As a whole, they tend to be very friendly, community oriented and proactive. They fought to save Eastern Market when chain grocery stores nearly went out of business, and are now fighting to save Market 5, a local theatre/arts center. The sense of community is very nice, something you don't generally get in DC.
Tons of places around here to try:
Montmarte. Great french food. 327 7th St
Tune Inn. The diviest of dive bars. 331 1/2 Pennsylvania Ave
Port City Java. Some of the best coffee in Southeast. 701 North Carolina Ave
Todays Pizza. Great pizza, sketchy place. 531 8th St
Market Lunch. Don't try breakfast. 225 7th St
Mr. Henry's. Nice outdoor drinking space. 601 Pennsylvania Ave
Pas Placitas. Delicious Mexican. 517 8th St
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Good luck with parking.
It is serviced by BWI, Andrews Airforce Base, IAD and DCA.
It is also serviced by the Metro's blue and orange lines.
Bus Routes:
Check Craig's List or sublet.com for great deals
Sure, especially with three parks and all those markets.
Obviously, Eastern Market is right there. It is on the National Register of Historic Places and has not only food, but a wide variety of goods for sale.
Several of the Smithsonian Institutions are located in or around Eastern Market.
Market 5 Gallery, which has art shows, theatre, etc.
2 Shakespeare theatres
There is a concert series here during the summer
Capitol
National Mall
Market Festival Arts and Crafts Fair
Flea Market
7th Street has some great stuff on it. And the area around the Market is always fun.
Grocery:
Obviously, Eastern Market
Marvelous Market
Mott's Market
Rolands
Fresh Food Market
Weisfeld's Market
Banks:
Washington Mutual
Citibank
State Farm
National Capital Bank of Washington
Gyms:
Curves
GI Janes Fitness Boot Camp
Tango
Everhart's Karate
Robert Brent Elementary School
St Peter's Interparish School
Hine Jr. High School
Apple Tree Early Learning
Capitol Hill Day School
Tyler Elementary School
Equal Housing Opportunity