I've extracted a tar file with sudo to / which contained the following file structure:

Since then I can't do anything on my server anymore, each command is either permission denied or command not found. Also most of the programs, like Apache or SSH, don't work anymore, although the system is still writing to log files and the virtual machines are still running.
At first I thought I've overwritten the whole /usr directory, but actually that's not the case. I've accessed my server via a rescue system and all the files are still there. The permissions seem to be okay too, they all belong to root with 0755 (see second screenshot). The new files have just been extracted into the existing folders, so I don't see how this could have done any harm (I've now deleted them).

Unfortunately I don't have a backup of /usr.
I really don't have an idea what to do now, so it would be great if you could help me. Please let me know if you need any more information.