Yes, Zipcar operates something very similar, but the drawback is that you can’t have a one-way rental. Actually Streetcar used to allow one-way rentals, but there was a massive £24 surcharge (because a staff-member had to pick it up and take it back to its original bay).
The idea is that the scheme takes private cars off the road, as it has done in most cities that similar schemes operate in. The danger is that it could pull people off the buses/trains and on to the road. But surveys by Zipcar/Streetcar have found that most of their members are either former car users, or people who were considering buying a car.
I think it’s still a very long way off in London. Car-sharing is already more developed here than it is in other European cities, and I can’t see Zipcar being too happy about facing a public-sector rival.