As explained at Quartz:
Public libraries are a relatively new phenomenon. Before the 1880s, when Andrew Carnegie started funding the more than 1,600 library buildings that bear his name, most libraries in America were subscription-based, with members funding and shaping the collections. As free public libraries sprouted up across the United States, membership libraries mostly died off, but 19 non-profit membership libraries still exist, and are reinventing themselves as cultural centers and the coolest coworking spaces you could dream of.I’m an itinerant freelance writer and work most of the year in Cincinnati, where I discovered the Mercantile Library a few years ago, when it was hosting a small concert. The majestic space immediately appealed to my aesthetics, but I had no idea of its history. The Mercantile was founded in 1835 as a young merchants’ association library—the collection is generalist, though early on there was a ban on novels that has since been reversed. The original cast iron and walnut newspaper reading desks remain in the building that the association secured with a 10,000-year lease for $1 annually.
Membership in the Mercantile Library is $55/year. Even with that incredible lease, there is no way that such dues can cover maintenance and staff expenses. They must have an immense endowment.
There is a list of other membership libraries at the link; check there to see if one is available near you.