I don't do off-season trips so am not sure what the restaurant situation will be like. San Sebastian is not overly blessed with major "attractions" in the traditional sense. I think the San Telmo Museum is probably the most important. You can check its website to see what its hours are in November. The fairly small historic district (which is where a lot of the restaurants and tapas bars are located) will of course be available for wandering.
San Sebastian is a geographically convenient base for visiting the coastal towns of Hondarribia (the largest of the ones I'm mentioning) and Zarautz/Getaria/Zumaia. Some are bus destinations; others can be reached by train. I am sort of doubtful about November in those places, especially the three smaller ones. They are little, historic, former fishing villages with restaurants/cafes on the water. I don't know how many of those will be open in November.
The fastest way to Bilbao (which has more museums, a larger historic district, and fewer tourists--at least in peak season) is by bus. I'm not the only person on the forum who preferred Bilbao to San Sebastian, but I am not a bar-hopper and S.S. is very popular. Bilbao is a larger city, much less tourist-driven, and will not be closed in November.
The interior capital of Vitoria-Gasteiz is another possible destination. It has a large medieval district on a hill and some museums that I'm sure will not be totally closed in November, because Vitoria-Gasteiz is not terribly tourist-focused. I think V-G is over an hour from both Bilbao and San Sebastian.
Bilbao's airport is considerably busier than San Sebastian's, so you may have more flight options there. I don't know the details, but I'm sure there's decent bus transportation from the Bilbao airport to San Sebastian. You can look for flights on skyscanner.com.
Basque Country weather tends to be cool and damp even in mid-summer, so you should be prepared for the possibility of a dearth of sunshine and the certainty of fairly short days.
