If you notice that PDFs in your library are no longer present and a missing PDF warning sign is displayed, the first thing to try is to go to Help > Troubleshooting > Repair file path metadata.
This repair method attempts to re-connect files that are orphaned on disk with corresponding publication entries in your library.
When you run this menu command, please note the method doesn't have any progress dialog yet, so it simply blocks Papers while it is running and you get that spinning wheel cursor. The method is done when the regular mouse cursor is available in Papers again. After the repair, the log in Console.app may contain some related warnings.
If that fails, perhaps you had moved the library or renamed any of the subfolders containing the library thus you may not have pointed Papers to look in the new location for the PDFs.
Try opening Papers in maintenance mode (start Papers with the ALT key held down. See screenshot), check that is is still pointing to the correct library and if not, choose to "Browse" to it.
If this still fails, you can restore from your Time Machine or Dropbox following these steps.