If the area is not dense enough, public transport will be difficult to organise. You would need some form of a hybrid between Uber and a bus.

Also, many professions require high levels of mobility.

Finally, unless you introduce some kind of permit system and create serious beaurocracy for enforcing it, people will just pretend to have their own companies and drive "company" cars anyway.

Nature will find a way :)

And I say this while taking 50% of my trips with public transit (the rest with Uber). I seriously hope we get rid of gas cars one way, but banning private ownership would be crazy.