Inside Takoma Park on New York Avenue. Residents can get free Community Borrower Cards which will enable you to check out books, but if you want to use their oniine resources, you need to do so from inside the library.
Also in Takoma Park, west of Flower between Maplewood and Carroll on the WAU campus. No community cards or access to resources, but it is open to the public and you should visit if you live in Takoma Park. The library is lovely.
A regional library in the Prince George's Library system. 6530 Adelphi Road near PG Plaza. As they have an exchange arrangement with other large regional systems, the PG card will be free if you are a Montgomery County or DC resident. This library houses an exceptional Maryland history collection. They are currently in a temporary location at 6502 America Blvd.
Near Takoma Park at 416 Cedar Street NW. Only a couple of blocks from the Metro station. DC is also part of the exchange system among the large, regional library systems. Many interesting programs, and if you are interested in e-books, get a card here. The Renaissance Revival building dates from 1911.
From Takoma Park it's just a hop, skip & jump to some renowned libraries. Take Metro to the Folger Library and the Library of Congress, both at the Capitol South stop (or walk up from Union Station.) The DAR Library is at 1776 D Street NW near the Elipse. All the major universities have extensive academic libraries. On this map every little red dot represents a library.