(an earlier version can be found at http://ping13.net/hacks/dropbox-and-webdav-old)
It's really easy, assuming you have your own server running somewhere. By the way, my hosting provider is Webfaction (affiliate link).
So, here we go:
Make a link from
~/Dropbox to your WebDav directory:
Dropbox -> /home/user/my-webdav-repository/Dropbox
Install a CLI version of Dropbox on your server. Also remember to install
dropbox.py, mentioned on the Linux Download page. (Update 2013-02-09: setting the
LANGUAGE environment variable seems to help people having permission issues, cf. https://twitter.com/waltonjones/status/237173959530012672)
Now you can read and write to Dropbox (referral link) without having Dropbox installed on your local machine. Of course I'd advise you to install the Dropbox client if possible, but there are situations you just can't: You are on a foreign machine or your are on an iOS device and your app does not support Dropbox, e.g. Pages.
If you don't have access to a server machine, you might try DropDav - I can't vouch for them as I didn't tried their service. They seem to be responsive, though.
If you want to sign up for Dropbox and want extra storage, you may use this link. If you follow the link, you do me a favour as I get extra storage as well.