Unfortunately, "no roads" is exactly what the OP wrote. I'd like to think he really meant either "no roads open to personal cars", or "no wide roads", or "no roads with street parking", because I think those are all fairly reasonable positions for a dense city, but I find it hard to interpret his comment that way, the way it's written.

But you're right: it is possible to have roads that aren't open to private vehicles; I've seen this in Amsterdam, for instance, where remote-control bollards will lower for certain vehicles (trash trucks, etc.).

Personally, as someone who lives in Tokyo, I think the easiest way to solve this problem is the way it's mostly solved here: simply eliminate all street parking, and require anyone living in the city who wants to own and keep a car there to prove they have a place to park it. We don't have roads banned to private cars, or congestion pricing (aside from tolls on the highways), but there's remarkably little car traffic for a city with this many people, and I think it's pretty clear why: if there's no place to park other than inconvenient and very expensive private parking lots, why get a car, or use it in the city if you have one? (To be clear, a lot of people actually do own them, but just don't drive them much because there's no place to park at the places inside the city where they want to go, so many cars are just for weekend use.)

There are relatively few cities (of which Manhattan is probably one) where, if I lived in the city, I wouldn't own a car but--yes--I might park somewhere on the periphery where it was cheaper and not use it day-to-day. I know very few people who aren't maybe just out of school who don't own cars even if they live in a city.

It is easier to get by with things like Zipcar and Uber but I'd still probably find it had value so I wasn't structuring things around not having a car.