Vehicle non-ownership is only cost effective in urban cores.
Even then, it's questionable. I have some vehicle-ownership-eschewing friends, but they're compensated by their employer, who spends a minimum of $20/day, or $400/mo, just for their commute. You can easily pay for a decent vehicle -- including fuel and maintenance -- for that much, let alone when you factor in the extra $100 to $200 they're paying for personal Uber rides and weekend car rentals so they can actually leave the city.
Self driving cars should make on-demand car renting far cheaper, and make non-ownership a viable option for far more people.
It will only be cheaper if those savings are actually passed along to the consumer rather than gobbled up as profit. I predict that a driverless taxi will be cheap for just long enough to make taxi driving obsolete, and then once that happens, prices will rebound to just below the cost of owning a seldom-used vehicle.
And there are some parts of vehicle ownership that are difficult to reproduce. Instant availability with no wait-time, a somewhat reasonable assurance that you're not sitting in or touching the remnants of someone else's bodily fluids, and a low penalty for forgetting your wallet or phone in your own car. The value of those features is, of course, subjective.