I have a desktop and laptop both with Ubuntu hardy 8. I mostly use the
laptop and home is getting too big to conveniently back up to cd.
Neither machine is currently a server. I access the internet via a
broadband modem router. Dynalink RTA1025W connected with Ethernet cables.
The modem router has a firewall and I have reset its login and password from the default and also turned off the wireless for reasons of security.

My question is: Is it possible to securely network the two machines through the modem router without making my desktop a server or would that make things easier?

If you have sshd enabled on your desktop and that is what you are wanting to back up to, I suggest rsync. It is very fast and can run over ssh. It also, once you get the initial sync set up, will only copy over changed files...which makes it much faster later on.

A simple example would then be:
This will do a verbose(v) archival(a) sync with compression (z) from /data/tmp to the host foo with user's credentials.

There are plenty of how to pages out there regarding rsync. I love it!