Hello,

Welcome to OSM! To mark an unused path as inaccessible, you can use "disused:highway=path". It will remove the path from the map and from the routers that use OSM data, but still keep it in the database for any future update. See https://wiki.openstreetmap.org/wiki/Key:disused:*

For old paths, it is a good way to map inaccessible paths, but don't worry, it is also OK to just delete them, if you are sure it does not exist anymore. 

Happy mapping!
juminet