The Spetses Museum occupies the mansion of Hadjiyanni Mexi, one of the island's revolutionary leaders. Its collections range from ancient ceramics and Roman sculpture to Byzantine icons, local dress, household objects, and material from the 1821 uprising.
It overlaps a little with the Bouboulina Museum, but the wider sweep is useful. Go here for the island's long history; go to Bouboulina's house for the human drama.