Depends on where you live and what you do outside. Over the years, I have ridden my bike to half a dozen different countries, starting from my doorstep. That might be rare, but it's definitely not "impossible without a car".
I also happen to have access to a very active market for skiing/snowboarding weekend trips where a bus takes you to an hotel in the mountains on Friday night, you spend two days on the slopes and get back on Sunday evening. There are all kinds of people on those buses and most of them arrive at the meeting point by car, so it's definitely not a choice made out of necessity.
effectively there are moderately common road/condition combinations that you cannot drive in a rental car.
Where is this? I live in northern part of Scandinavia, and we manage just fine with normal winter tires. Snow chains are pretty much only for semi trucks. Most people even use studless winter tires these days.
And of course rental cars have winter tires in the winter season, anything else would be illegal.
The US does not mandate snow tires at a national level, and in many places (even northern states where snow and ice are common) tire chains are illegal because they can damage the roads.
If you're going up steep hills covered in ice, winter tires or even studded tires are not going to work. You need chains. Then again, the best rally drivers are Scandinavian, so maybe you guys are just better drivers than we are :)
Lots of them in the West. Little Cottonwood Canyon (at a minimum) in Utah. Tahoe from the Bay area (Donner Pass) under certain conditions http://www.dot.ca.gov/trafficops/trucks/chains.html. In a number of cases you can get exceptions for approved winter tires but it varies.
Zipcars with ski racks might be a little hard to find in Seattle during the winter, but they do exist.
If you stay in the city almost all the time it's probably great. The minute you want to do something outside of the city it's impossible without a car.