Nope, but if you're interested in fitness there's a great path that connects Oia and Fira. It takes a couple hours each way but gives some spectacular views of the Caldera!
No, but you can run or walk up and down the stairs to the harbor!
I'm sorry I dont know. We were so busy walking and sight seeing everywhere that I don't think we thought of going to the gym!
No there isn't a gym.
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