It's entirely possible to live without a car - very, very much so, but be prepared to pay if that's the case (i.e. neighborhoods well served by transit are expensive, think Manhattan rents... up to $2K for a studio? You bet!).
I'm really not much better off here in NYC though because I live way out on a point in Queens. The nearest public transportation is a bus that's over two miles from where I'm living. The long and short of it is I commute two hours each way every single day.
It's a certain kind of hell I'm not long to suffer.
But I guess my biggest challenge at this point is not only do I not have a car I don't have a license; never bothered and now I regret it. I'm not alone in NYC in this regard. My coworker has lived in NYC all his life and never had a license. He's in his mid-thirties. So I'm trying to figure out if it's practical to attempt the move to SF. Of course there's many things to figure out besides the logistics... but certainly this thread is very helpful in regards to getting around the city.
And re: expensive neighborhoods for transit. I think your numbers are quite a bit off - sure, if you want to live near the main Caltrain station, you are looking at that much. But you can easily find a large 1 bedroom for $1600 within 5 minutes of the light rail or BART for a total transit time to downtown of 10-15 minutes.
Being near BART is going to set you back at least $1700 at the very low end for a studio, $2000 for a 1BR. More for both depending on quality of the place, naturally. You might hack 10% off either of those prices if you pick up a particularly sweet deal.
FWIW, I signed my current place last month, so that's when my knowledge of the current rental market ends. I talked to numerous people including an apartment hunter, all of whom expressed incredulity at how much prices are shifting, even on the order of a couple of months.
General San Franciscan rule: If your neighborhood has parking, it isn't hip.