In this on a PC or a console? If it's Windows check your "Documents\My Games\FINAL FANTASY XIV - A Realm Reborn folder" as the game needs to write files inside that folder (specifically FFXIV_CHR folders within it, one for each character you have). If there disk is full and/or the game doesn't have file permissions to write there, you'd get the error you're experiencing.

Possibly the files were created originally on a different computer/user account? If so you could just delete the whole folder and let the game recreate it, but this will wipe out all your settings/hotbars so if you have anything you want to keep there you'll have to manually check/adjust the security on the folder.

If it's on a console though then I'm not sure sorry (other than just check the disk isn't full, the character files aren't a lot of space though and I would imagine if your console ran out of space you'd have bigger issues)