Ok, it’s a tiny island, we know that, but there are a lot of things for most non vehicle interests humans to do.
If you are a swim-beach guy you can find some beautiful beaches on the island. The famous one is Tiganakia followed by Patelia, Limnari and Piso Patelia. All beaches are reachable by feet, we only have two cars and one motorbike on the island. So don’t even loose a singe thought of renting something here. The water is clear and the beaches are natural intact and virgin. So please keep it like that. It is also possible to organize a boat with a local, to visit Aspronisi (The white Island) or the Island Marathi, which is a 15 minutes ride, and has a beautiful beach as well.
For the tracking freaks, the island offers an old italian fortress and a cave, both like to be discovered. The way to it is already impressive. You will pass by strange air shaped olive trees with an amazing view of the surrounding Islands - Patmos, Samos, Agathonisi, Lipsos and the Lazaros house. Lazaros who? The best Greek dancer on the Island.
Also very popular is the "doing nothing day", the “my 350 page book read in one day” day, the “snorkelling, fishing and philosophy” day.
I guess there are a lot of other activities you also could do...