Some services offer domains for free, it usually requires an appending with their domain name to yours though, so for example:
www.somethingcool.no-ip.org might be your domain. Check out:
www.noip.com
www.dyndns.org
www.zoneedit.com
www.changeip.com (.org?)
For more info.
Short of that, no you can't just "make up" one
There would really be no point in having registrars if that were true
You can however, for your local LAN choose whatever you'd like, but realize that will only hold true in your local lan, and need to be setup in /etc/hosts or your DNS software locally to work.
Cool
