Q:

How did the Greeks keep a prytaneum alight in storms?

In ancient Greece, prytaneums were present in every city, village and state. They were lit with fire from the place in which the people settling came from. It was tended to by a leader of the establishment and was never allowed to be fully extinguished. And if the sacred fire was allowed to die, the city, village or state had to rekindle the fire from the prytaneum of the mother or parent city, state or village.
In a storm, how did the Greeks keep the fire lit?

A:

I'm not quite sure why you think this would be a problem. A Prytaneum was a building, with a roof, and presumably doors that could be closed and windows that could be shuttered. While modern westerners tend not to have open fires indoors, the methods of keeping a fire alight indoors during rain without suffocating had been established for thousands of years. 

