Pro-tip for life: Don't tell the Internet it can't do something. Case in point, Nintendo shut down its official servers for Wii and DS online play just three days ago, and already there's a community running their own service in its stead. While it may take some extra work and tweaking on your end to configure a DS or Wii system to be compatible with the unofficial server, it is theoretically possible to still get in a race or two of Mario Kart.
We say "theoretically" because, being unofficial and all, the server isn't as stable or reliable as Nintendo's was. Not every game with online play has been tested for compatibility, and even those that have been tested could crash under server stress. Still, the project's wiki page notes that it is being actively developed, so it's possible things will improve. In the meantime, wear a hard hat and watch for falling goombas.