The Nintendo Gamecube is fully capable of online gameplay. The reason Nintendo does not publish online games, according to them, is because they have not yet worked out a satisfactory business model for online games. Frankly, I agree with them here. I love online games, but I would never want to pay a monthly fee, in addition to what I already pay simply to get internet access, to be able to play online games. There are some online games for which the developers provide free servers (such as games played through Blizzard's battle.net), and I play these games, and greatly enjoy them. But I would never play X-Box Live. And I don't expect all companies to follow Blizzard's example and provide free servers.
By the way, Mario Kart was a poor choice of examples. While Mario Kart is not an online game, it can be played over a LAN, in which case eight people CAN play at the same time.