That you can still view the file system at all is promising as it means the drive is still working. If the data is at all important, immediately save a binary image of the entire drive. Something like
diskimager will work. You can later work with that image if the drive itself become inaccessible.
More information about the OS you're using and how you're viewing permissions on this drive might help. exFAT normally doesn't implement file permissions and should be readable on just about anything with support for the file system. You can set ACLs on exFAT volumes, which might be locking you out, but it should be possible to
override those if that is the case. That said, if a bad ACL is the issue, just plugging the drive into a different system might make it accessible.
If the file system is just corrupted, running basic recovery software like Recuva should be able to recover your files and since there should be no security metadata attached to the files themselves, they should be immediately accessible. Things should only be difficult if the hardware has failed or something has encrypted the drive.
