Search your conf/httpd.conf file for this exact string: <Files ~>. If you find it, that's your problem -- that particular <Files> container is malformed. Delete it or replace it with <Files ~ "^\.ht"> and restart your server and things should work as expected.

This error appears to be caused by a problem with the version of linuxconf distributed with Redhat 6.x. It may reappear if you use linuxconf again.

If you don't find this string, check out the previous question.