My old 2006 eMac computer is broken and the monitor seem to be the broken part. The computer will still boot up. With a little direction and encouragement from my brother, I logged into the Apple eMac computer remotely over my home LAN from an Ubuntu Studio laptop. I am now copying the Library/Mail files off to a network drive. I have done this logging in remotely to a computer with a dead screen, once before for a friend whose computer was broken. A Linux computer is a good option for logging in remotely. Then I simply copy files off the hard drive to a network drive.
Tomorrow I pick up a brand new Mac Mini from the local Apple Store and bring it home and try to load the old emails into the new machine. I then may try to export those old 1995 to 2010 emails to Thunderbird.