This is where not owning a car can be a bit of a pain and/or expensive if you don't do it right. I use zipcar for local quick trips (their hourly price includes insurance and gas, and their service has been great), but at $100+/day it doesn't work too well for going out of town for the weekend or a week. For those trips I do a normal car rental. American Express cards (at least the basic ones I've had) will cover damage to the rental car itself, but don't cover the liability half. For that you need a non-owners liability policy, for which State Farm is the only company I've found that does that. I've heard USAA bundles it in with some of their other things like renters insurance, but I don't have USAA to verify.
So it's a bit of a juggling act, and if you use a car a lot it can be expensive. Sometimes I'll go through spouts of car rentals that make me question not just owning, but then there are times like 2015 so far that make up for it, where I've yet to use a car at all this year.