Relative to Piedmont, Orinda, or Danville, a lot of the East Bay is attainable. Relative to a national average, less of it is. The honest map of more attainable homes runs through Antioch, Pittsburg, Bay Point, Oakley, parts of Richmond and San Pablo, Hayward, Ashland, and Cherryland.
Attainable should not mean unexamined. Ask about the commute you will really do, the flood and industry context on the shoreline and the Delta, and the block, not the city slogan.
If your budget is higher but not luxury, look at Concord, San Leandro, Union City, Newark, and parts of Oakland outside the famous restaurant streets. Oakland in particular is many markets sharing a name.
Towns
Delta & East County
Antioch
A Delta city at the end of the BART line.
OpenDelta & East County
Pittsburg
A river city with BART and a downtown trying to face the water.
OpenDelta & East County
Bay Point
An unincorporated community at a BART station.
OpenDelta & East County
Oakley
A Delta-side city between Antioch and Brentwood.
OpenSouthern Alameda
Hayward
A big, varied city between the shoreline and the hills.
OpenWest Contra Costa
Richmond
A working waterfront with a village hiding at Point Richmond.
OpenWest Contra Costa
San Pablo
A compact city between Richmond and the hills.
OpenSouthern Alameda
Ashland
An unincorporated community between San Leandro and Hayward.
Open