Here's some more suggestions (including some links for those already mentioned):
Self Storage Millbrae
Safe Harbor - Napa
Eagle Rock - Napa
Security Storage - Napa
Subterranean Wine Storage - Oakland
Wine Bank - Menlo Park
The rest are in San Fransisco. Hope that's at least a little helpful.
Yes there is quite a history of wine storage up and down the Peninsula, and Subterraneum in Oakland which RL cited. Though I understand the OP's wish for something nearer home. I don't know Tom's location but if it is somewhere like Pleasanton, a drive from there to Menlo Park corresponds to the travel between some countries in Europe.
Before the owner built up Wine Bank deliberately for this purpose a dozen years ago, he spent time studying and visiting consumer wine storage facilities in the US region with the greatest use of them, Los Angeles. I think it's an impressive and classy firm. If a bit remote from the East Bay, except for locations near the Hayward - San Mateo bridge.