There are only two steps to turning them back on. Rename the ht.access file in the MODX root to .htaccess and revert that friendly_urls System Setting to yes (then manually delete all files in the core/cache directory before testing).
Now that you've determined that the site works with FURLS off, if it doesn't work with FURLS on, the problem must be either in your .htaccess file (usually the Rewrite Base line), or a configuration problem at the host (usually because the Rewrite Engine is disabled or the host has things set up to ignore .htaccess).
Stuff like this is exactly why I always go with
MODX Friendly Hosts. They usually don't cost any more, and even if they did, my time is worth too much to mess with a host that makes you jump through frustrating hoops to run MODX.