This excellent museum, set in a grand 1891 red-brick building that once served as the Custom House, covers Key West's history. Highlights are the archival footage from the building of the ambitious Overseas Hwy (and the hurricane that killed 400 people), a model of the ill-fated USS Maine (sunk during the Spanish–American War), exhibitions on the role of the navy (once the largest employer in Key West) and the "wreckers" of Key West, who scavenged sunken treasure ships.
There's also info on rum-running to Havana during the Prohibition days and some excellent folk art: seek out Mario Sanchez' art naïf paintings of Key West from the 1960s and '70s.
