TheOutcaste has some good tips. By the time AVG and Avast release their final definitions for these systems, you should be fine, if you're really concerned. No one really writes viruses for these antiquated systems anymore. While these legacy systems are still very useful to me and others, they are in the minority compared to the installed base of 2000/XP/Vista. Viruswriters are just learning how to infect Windows 7 after all LOL
Also, I think it's highly unlikely that you will get viruses with Windows ME. It is a DOS-based system along with 95/98/98SE; although Microsoft did disable DOS mode at boot-up for ME to speed the boot process, it is possible to re-enable it.
I have been running 95, NT 3.x, 3.5x, and NT 4.0 on my old boxes for years without virus protection. I've never gotten viruses. Basically, if you are careful about opening attachments only from know sources you
should be fine. I do, however, run Outpost Firewall on these old boxes. You can always be hacked without a firewall, and OF has the lowest system requirements without requiring particular service packs, etc.