Doing "View Source" on any page (when they fail) shows no content except this:
<!--include virtual="/${GD_ERROR_DOC}" -->
That's not anything to do with MODX, but I don't know what it is.
If there is a core/cache/ directory, try manually deleting all files in it (be careful not to delete the whole core directory -- you just want to clear the cache). If there isn't a core directory, you're using MODX Evolution rather than MODX Revolution.
If that doesn't work, one temporary solution (based on mmcgee's discovery) might be to open the database in PhpMyAdmin and find the modx_system_settings table. In that, there will be a setting with 'friendly_urls' in the key field. Change the 'value' field from 1 to 0. Then clear the core/cache directory. If that fixes things, it's definitely a Friendly URL problem and the problem is almost certainly in the .htaccess file in the MODX root. If you look at that file, it should be clear that it's a MODX file. Maybe your host has overwritten it (or some hacker has).
Another possibility is that mod_rewrite has somehow gotten turned off on the server.
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