Can anyone recommend an affordable hotel in Washington DC that is close to all the major attractions?

Crystal City is definitely a good location in Northern Virginia to stay. There are numerous hotels including Hyatt, two Marriott, Holiday Inn, etc.which all have weekend rates under $100 per night. The Marriott's connect to the Metro station via an underground shopping mall, which makes it especially convenient in bad weather. Crystal City is located right on the Metro Line which is served by both the Yellow Line and the Blue Lines, which would give you easy access to the locations you mentioned, as the Blue Line stops at Arlington Cemetery and in the middle of the National Mall, for easy access to all the Smithsonian museums, while the Yellow Line can take you directly to the National Archives and Chinatown.