When I try to check the box turn on iCloud for the Photos App in System Preferences, it gives me a spinny wheel and says 'checking library...' for days. What does this mean? How do I solve it?

Restart the Mac, then try to open Photos directly. Can you open your Photos Library in Photos? Can Photos show all photos in the library and can you edit them?

Where is the local copy of your Photos Library stored?

Photos cannot work with a library, if it is on a network share or on a volume used for Time Machine backups. Photos on Catalina has also problems with referenced files.

Did you just upgrade to Catalina? Did you previously upload all photos to iCloud Photos and are all your Photos already in iCloud?