As previously stated 20002 and 20003 are Capitol Hill zip codes, but the neighborhoods change within those zip codes. Capitol Hill is primarily the 5 or 6 city blocks closest to the Capitol building. Capitol Hill East starts around Eastern Market at 7th Street and extends all the way to the Anacostia River. Within that is the Eastern Market neighborhood, Barney Circle which is around the Congressional Cemetery (named for where the old trolley car system used to have a turnaround/circle), Klingman Park/DC Armory which are north of East Capitol Street and extends up to the H Street neighborhood/Atlas District. South of Eastern Market is the Navy Yards area which extends to the I395 border.
If you want to see the breakdown of all zip codes in DC, or anywhere, this is a great website to use: www.hipcodes.com