I lived without a car in Chicago for three years with absolutely no problems.
Also, there is a spectrum from being totally car-dependent to not owning a car. My parents live in the D.C. suburbs and they have more cars in their garage than people who live there. Because you can't even go to the drug store without driving several miles (and this isn't a far-flung exurb, they're only 17 miles from downtown) my brother and I each got a car as soon as we could drive. Meanwhile, I now live in Westchester (a suburb of NYC), and while I own a car I only drive it a couple of times a month (Costco trips mostly). The same is true for Chicago. Unlike in Manhattan, most people do own cars, but a large number of people don't use them that heavily because the suburbs are easily accessible by train.