Do we have any idea how big the island of Dragonstone is?

Pretty damn big. Dragonstone is a volcanic island, and the “Dragonmont” volcano was big enough to provide lairs for six dragons. The island has a harbor with docks extensive enough to hold two hundred ships. the port has multiple inns, there are multiple villages on the island 

Eyeballing it on the map, it seems to be about 50 miles long and maybe 12 miles across.