I am trying to run chown on a directory and I get a permission denied. I have tried running sudo and also running the command as root.
If root doesn't have permission, how can I change the owner of this directory?
It sounds like you are trying this on a read-only filesystem. (mounted with option ro, or some other way) If that is the case, there is no way you can achieve what you are trying other than changing the read/write status of the filesystem in question.
With more details we can help you more:
at a minimum, look at the full path name and see how your filesystems are mounted