The US is more difficult, at least partly because of the distances involved, but it wouldn't be impossible in the UK to imagine a situation where public transport can take over most normal journeys. Most of the country will never be as well-provided as London is at the moment - but judicious re-openings of branch railways (maybe as light rail or even dedicated bus routes) could go a long way towards reducing most people's reliance on the car.
You couldn't outright "ban" private car ownership - particularly in rural areas such as the Scottish highlands or mid Wales - but most people living in urban or sub-urban environments could probably be provided for with well-planned improvements to public transport, and it's possible that a fairly easy reduction from one-car-per-adult to one-car-per-household could be made.
There would be other complications of course. My current 45-mile (one way) commute, for example, would take me at least two hours by public transport (20 minutes on the bus, 90 minutes on two trains, 10 minute brisk walk) and cost just under £25 at peak time (one way). It would mean I wasn't as flexible as I am at the moment about starting early or working late. A re-opened branch line might knock 10 minutes off that journey as it so happens I live near a closed line which would go more directly to the town's station than the bus does, but realistically I would be looking for a new job closer to home.
My wife could get to her "base" quite easily by public transport - though it would take her nearer an hour than the 20 minutes it takes to drive. She does a lot of work "in the community", some days travelling 50 or 60 miles from base, but this could potentially be handled if her work provided a pool car, though with current mainstream technology a battery car would be marginal.
We live in a semi-rural location and I don't think we could manage without a car at all (there's only so much shopping you can do online), but down to one car could have potential given some changes to public transport...