I don't think our poster is expecting to cover northern Spain from the Basque Country to Galicia by slow train from one base. That would, indeed, be impossible. I believe he (?) just meant they want to pick a place somewhere in that part of Spain and see as much as they can.
I agree with Isabel that San Sebastian would be a practical option. I prefer Bilbao as a city (much less touristy, much larger historic district), but I think the side-trip options may be better from San Sebastian. Those could include Hondarribia, Zarautz/Getaria/Zumaia, Bilbao (might be worth two trips), Vitoria-Gasteiz, St Jean-de-Luz (France), and Bayonne (France). I wasn't terribly fond of Biarritz (also France), but it is also doable. I've read that Lekeitio (Spain) is nice but haven't been there.
It might be useful to know that bus service between San Sebastian and Bilbao is quite frequent, but it's worth traveling once on the very slow train between the two. The scenery is good, and you feel like you're in a tourist-free zone as you travel through that part of the Basque Country.
Alternatively, you could put together a week-long stay in one place in Galicia and have a very nice time, but I feel as if there's more variety to be had by basing in San Sebastian. Santiago de Compostela would be the most interesting base in Galicia, but I'm not sure how the transportation would work out from that single base. I also stayed in Pontevedra and A Coruna and did some of my day-tripping from each of the three cities.