Sunol Regional Wilderness
It's really not very far from the McSprawl which characterizes the Bay Area, though relying on public transit won't get you near this place in the least. So seat yourself in or on a vehicle, follow Calaveras and Geary Roads, and end up in the oak forest at Sunol Regional Wilderness.
"Wilderness" is a romantic flourish: the area is parkland which is grazed to the roots by cattle, and hiked by loving crowds of people and dogs. My vision of wilderness doesn't include tidy picnic areas and clean toilets, but I'll live with the cognitive dissonance. Walking around the Wilderness is a ready escape from a too-paved Air Conditioned Nightmare, especially early in the morning. Others seem to agree-- enter the main parking lot at, Gawd, 7.30 on a Saturday morning, and, you know, there are other people already there!