I'm not sure Home Edition is up to that. Start by totally ignoring the Homegroup facility - that has absolutely no chance. In more conventional networking, you may need to eliminate the need for every system in the Domain to have only password protected accounts. This you do in Control Panel>Network and Sharing Centre, then restart the system and see if it will join the domain.
I have a feeling Home can only join Workgroups but I'd like to be proved wrong.
A Home Edition cannot authenticate with a 2003 domain, but you can certainly access resources on the domain if you have an account there. Just make sure that your user account has the same name and password on both systems and you can use the "net use" command to connect to a shared folder. E.g.
net use T: \\windows2008.mydomain.org\Home\ian
If the username is different you can supply the credentials when you connect by using the /USER: switch and specifying a password.
Home editions aren't designed with domains in mind... hence the Home designation. Very few people run a server at home... and probably even fewer bother setting up a domain. You need Pro, Enterprise or Ultimate versions to join a domain. However, as ijack correctly points out, you can still access domain resources.