You can go hiking in the city and up the mountains.
Or you can sit and contemplate life. But you probably want to do more.
There is always swimming. It's not a super island for beaches, check out the sand beach east at Mandraki. West, you just find this.
Pebble beaches, mostly. small sites. A water taxi can take you farther to some larger beaches.
Hydra's museums are smaller than the guidebooks make them out to be. Fewer things are open than you might expect. At least one of the old captain's mansions is now a museum of folk arts.
Village women used to have distinctive formal costume with variations for each village. The model on your right has money on a chain around her neck, showing off the wealth of her family - and calling attention to the size of her dowry.
Oh yes, eating. For some reason, a lot of people like to do this.
At times, it can be easy to think that the harbor is the island, but there is more. I've really not gone very far into the interior, and it is (by tourists) relatively unexplored.