The eMac was an all-in-one G4-based Mac designed for the education market and released in April 2002. It was the last Mac to use a CRT display and was sold at a low price to schools and other institutions.
First thing you should do is just try any other ethernet cable, to rule out a bad cable. Ether cables are fairly delicate and shouldn't be bent too sharply. Are you seeing "No cable connected" in the Network control panel? If you still get this message with a different, known-good cable, then your Ether port might be toast, but first you should try zapping the PRAM, resetting the NVRAM, and booting to a different system, to rule out software.
Of course, first check ifixit to see if they have something like it!
Zap PRAM: reboot while holding down command, option, P, R. Keep holding down till you hear 3 chimes.
Reset NVRAM: hold down command, option, the letter O, and F. You'll get a while screen with black text. Type "reset-nvram" and Return. Type "reset-all" at which point the computer will reboot automatically.
A cross-over cable would be used to connect like devices (eg. 2 hosts, a desktop to a laptop). Straight-through cables are used to connect different devices, such as a host to a router.
You may want to check up on your answers before posting.
Many recent Macs, including all Intel Macs, do not require crossover cables. See http://support.apple.com/kb/HT2274. As for this eMac, if it's ATI graphics or later, it does not require a crossover cable; they have Auto-MDIX, which automatically configures the ethernet connection. The OP does not say what they are trying to connect to, as you correctly point out, but one can presume it's a hub, switch, or router, as that's the most common scenario.