I have backed up the eMac's Desktop folder now. I am just verifying with the Get Info function that the two Desktops, the real one and the back up copy are about the same size.
Then I will continue the copying of files and folders from the Documents folder on the eMac to the back up copy on the network drive. Then I will back up e-mail. I will not bother to back up videos, photos, music or applications this time, as these do not change much. These especially do not change much on the eMac.
Back in early 2005 I had only an eMac and a Thinkpad. My philosophy was to keep all e-mail on the eMac when I got a Macbook to replace the Thinkpad that had broken that year. In the last six months I have lifted this policy a bit and generally read my e-mail on the Macbook. I did not want to do this for security reasons; given that I might loose a computer like a laptop outside the house I thought it better not to store e-mail on it. But I leave the e-mail on the server with the Macbook so it is still all downloaded eventually and stored on the eMac. But the eMac is getting old now. Also I am having trouble these days backing up the e-mail folders on the eMac to external drives and even more trouble backing up to DVD. When I did my e-mail on the Thinkpad before I started using only the eMac for e-mail, I could back up the e-mail to DVD no problem from the Thinkpad. There was simply less e-mail to backup back then.