Can't you copy the files as root?
However, if the filesystem is in trouble, I'm not sure that operating system commands would be safe to execute.
The safer way to proceed is to keep a clone copy of the partition in question (by dd command, clonezilla or other software), before any repair activity. Then you can retry recovery/repair work in case of unsuccessful result.
If you open the partition as root, you can copy the files to any other folder you have opened as root too. Please look in this discussion from Linux mint forums.
This is valid for "healthy" filesystems.
Furthermore, you could try to use the "photorec" free open source software to look for specific file types, in this case the content type of the files (are they html files? ). The problem with photorec is that it returns many files without their original names and directory tree structure, so you have to check them in order to find the files you want. Anyway, in a broken filesystem there is always probability to loose files or get some corrupted.
*** It is highly recommended to backup any important files before doing resize/move operations. ***