Hardware part would be relatively simple. That ethernet module looks like it would connect almost directly to the expansion port. The software would be the hard part. You would need a tcp stack which is a serious undertaking. The code might be ported from other small computer projects like the ethernet for Apple IIe, or the Contiki link.
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Teleplay NES modem was developed and multisystem games are definitely possible. Another way to do multiplayer games is to have the server just send out commands to the clients, instead of sending out graphics/sound data. That way each client does the graphics/sound processing using the commands and the server does not carry the load of every player. The more exact timing you need, the harder it gets to keep everyone synced. Turn based games for the NES would work best.