But it is Google Docs... Meaning I still have to use some other tool to have offline access (and backup) of my documents?

Isn't whole point of Dropbox to have offline access and (multiple) offline backups of your files?

those files will be opened with google docs web app. the web app itself can work offline. so your docs work offline.

How do I enable this feature? Whenever I try to access my docs offline the page simply does not load.

I am using a google apps account if that makes a difference.

There is an option in settings drop down from top left of the homepage od docs. Also the offline url is https://docs.google.com/offline. But you need to set up offline first from the menu i wrote about above.