The only place in the USA that I know of is Mackinac (pronounced Mack-i-naw) Island. Private cars are not allowed, although delivery vehicles and emergency vehicles are. Most people travel by foot or by bicycle, although horses are common as well. In the winter, people also ride snow mobiles around in the streets.
It is unlikely you could move there permanently unless you are rich. The island is small, and most of the money is generated from tourism. Real estate on the island is very expensive.
http://en.wikipedia.org/wiki/Mackinac_Island