It would also help if you told us something about your Internet connection. Cable or DSL? Which provider? Are you connecting through a router of some sort or do you just plug directly into the DSL/cable modem?
Meanwhile, here's stuff to try...
1) Type ifconfig eth0 -- what shows up? If you get "device not recognized" or a similar error your ethernet card hasn't been detected. You'll probably just have to modprobe the appropriate driver.
2) If the card is recoognized, check if ifconfig shows a valid IP address. If not and you get one via DHCP try typing "dhclient eth0" or "dhcpcd eth0" (not sure which one RH8 uses) and see if you get a good IP address and can connect.
Also, you should really consider upgrading to Fedora Core. RH8 is old and even Fedora Legacy has dropped support due to lack of demand. If you're computer is connected directly to the Internet with no firewall, please apply all the relevant security patches to your system and configure the firewall to block incoming traffic.