Where is Steam writing to and where is the drive mounted to? If the drive is mounted at /mnt/games, create the folder: /mnt/game/steam, run "chmod /mnt/game/steam" and then have Steam create it's library within there. So in the end, Steam's folder will probably be at /mnt/games/steam/SteamLibrary. This way it shouldn't matter what Steam is setting the permissions to.

You either have broken permissions, or a broken drive.

Try going into a terminal as your user, not root, and try creating a file in the directory you want to use for your Steam games. touch filename can be used to create files. If it spits back an error about permissions, you likely need to change ownership of the directory to your user. This would be sudo chown -R username:username /path/to/games/directory Make sure to only run this on a directory that's not used for anything else, as it can break system components if you run it on the wrong files.

If, when creating a file, you instead get a "Read only filesystem" or similar error, your external drive might be broken. This would be a whole separate thing to troubleshoot, and you'll have to look up your specific error for further actions.