Virtually every city and town in America has a public library that is supported by tax money.But could there be a PRIVATE library-one supported by membership fees and able to make its own rules and regulations.
@OliverM The exact term for what you are describing is a membership library, though. Those also exist. Private libraries are usually what you would refer to a University library as, or it's established specifically for a very limited number of people- such as a specific area of research of study.
Private libraries exist now, and were very common in the past. Many small town libraries in the northeast are named [Name of town] Free Library to indicate that they are open to everyone and are free to use.