Most people I know in the city have abandoned their cars. Zipcar has good penetration there but most people I know stick to public transit. I have friends that regularly do the trek from the Duboce Triangle to SF State without cars.
Public transit isn't as good as NYC or Chicago, but it's decent.
If you stay in the (miniscule) center of the city, yes. As soon as you take a job outside the city (say, in San Mateo), drop "easily" from the description. It's always doable, in any US city, to live without a car. But it's not always comfortable. SFBay is not comfortable without a car.
If you live and work near Caltrain or BART stations it's very easy to live in the city and work in the valley or east bay, or vice versa (though if you live outside the city you'll want a car).
Or if you're fortunate enough to work somewhere that has a shuttle that's another option. Many of the big tech companies do now, thanks to Google's lead. Off the top of my head: Google, Facebook, Apple, Genentech (I actually knew someone who would take the Genentech bus to work even though they didn't work there. Guess they don't check)
Of the people I know who live in SF and work in the valley, I'd say roughly half drive, and half take the train or a shuttle.
I actually know someone who moved to San Mateo from SF because it was easier to commute to his job in San Francisco from it. Muni sucks compared to BART, so moving Sunset->San Mateo was a commute upgrade (he works in the financial district). Even much of the East Bay is closer to downtown SF in commute-minutes than western SF is.
Public transit isn't as good as NYC or Chicago, but it's decent.