LibraryThing provides this option and also has a product called LibraryThing for Libraries. Evergreen and other open source are, of course, free, it just depends how much expertise you have.
A library could use LibraryThing to catalog and maintain their collection. It would be for a very small library and a small collection, but it has been done and LibraryThing would help with that.
When you are looking for products that don't require downloading of anything, that gets into Software as a Service (or the "cloud" as it is popularly called). Many library ILS vendors offer this, Polaris, Innovative, even OCLC are good examples. They host everything for you. This might be out of your price range for a small library though.
There are also Open Source Products that you can download and set-up such as Equinox/Evergreen. The company can provide some support, but you have to pay developers to customize it to your need. There are hidden personnel costs in programming, but it is still cheaper than any of the larger ILS Vendors.
I guess it depends on how small a library it is; LibraryThing for Smallest, Open Source, then bigger vendors.