To be fair, when you include BART's ridership (379,000), the Bay Area beats Chicago. You might also want to include the Capitol Corridor and ACE lines, which have minuscule ridership (5300 and 3700 daily, respectively), but add considerable trackage (168 and 84 miles). San Francisco's Muni Metro is a hair under 37 miles, but transports 162,000 riders daily. There's also San Jose's LRT, but let's leave that out for now.

A comparable rail map for the San Francisco Bay Area: http://www.urbanrail.net/am/snfr/sf-bay-area-map.htm

That puts the Bay Area commuter rail at 472 track miles and 598,000 passengers daily. Rather closer to your Chicago numbers.

But we've got to add in Chicago's CTA 'L line (709,000 daily riders, 108 miles). Tips lead to Chicago again.

Chicago benefits greatly from having grown up around (and not torn down much of its) commuter rail lines, including significant urban and suburban growth from 1860 - 1910, pre-automobile. It's occasionally noted in both areas that commute times a century ago were comparable to, or faster than, today (dedicated rights of way, less traffic, arguably fewer safety restrictions). The Bay Area notably tore down its old Key System commute rail, and several counties (San Mateo, Marin) have fought BART extensions for a half century, and saw much of its growth in the car-friendly post-WWII era.

The Bay Area's layout is actually fairly rail-friendly, with a few major transportation corridors (Peninsula, East Bayshore, CA-24, I-580, I-80, US 101 north of SF), though overall densities are low. Chicago has experienced a great deal of sprawl in the past 30 years, as well as employer flight to the suburbs, greatly increasing suburb-to-suburb commuting, though both SF and Chicago have also seen an increased appeal of the inner urban core especially for younger professionals.