All the Smithsonian museums and the zoo are closed. As well as the outdoor places already mentioned, the Capitol, the Library of Congress and the Supreme Court are still open to visitors.
Beyond the Mall, there are also free or cheap places that are not supported by the federal government, so they're not affected by the shutdown. There are many universities in DC; some have attractive campuses, with art exhibits or interesting lectures or performances. For religious architecture/culture/history/worship, there's the National Cathedral, the National Shrine of the Immaculate Conception, the historic Sixth & I synagogue, and the Franciscan Monastery. There's the Folger Shakespeare Library (actually pretty close to the Mall).
And of course, private paid attractions are still open--the Spy Museum, the Newseum, Hillwood Estate, etc.
