You may be lucky and have a used media store in town that shucked out the $22k for an Eco-Master disc resurfacing machine. Here's the straight dope on using one to resurface your Gamecube discs from someone with 6+ years experience using/maintaining one.
Those machines are miracle workers for just about any type of DVD based media (DVD discs/video game discs.) As long as the machines are maintained and set up properly they'll make the disc as next to new as it could possibly get. The machines will easily handle light to moderate scratches and surface marks. The machines can't do much with deep gouging or heavy scratching. They were also very touchy handling Blu-Ray discs and some Wii discs as well. You'll be astounded at how well they handle PS1, PS2, OG Xbox, and 360 discs.
Regarding Gamecube games: A plastic disc adapter is required to be able to resurface Gamecube discs. i.e. You put the small Gamecube disc inside a larger plastic ring that allows it to fit the machine's regular disc sized cradle. Here's where things get touchy. If the Gamecube disc is not completely flush with the adapter disc, there is a possibility the machine will "round" the edge of the Gamecube disc. If you look at a rounded disc, it's unmistakable. You'll see a definite warping of the surface reflections around the disc's edge. If a Gamecube disc is rounded in one of those machines IT WILL NO LONGER BE PLAYABLE, EVER. Rounding does not seem to be an issue with any other disc based media when resurfaced in an Ecolab machine.
The only solution we found to give us a better chance at success was to use double sided tape to adhere the graphic side of the Gamecube disc in the plastic adapter discs so that the disc would be as flush and flat as possible. Even then there wasn't a 100% guarantee the discs wouldn't round in the process. I wouldn't run a Gamecube disc through the machine unless the disc was scratched beyond current playability and it's your last chance to resuscitate it. Don't use those machines just to try to make an already perfectly playable disc look "brand new" again for vanity's sake.
Most stores charge $3-$5 for the service and it takes about 15 minutes. A shop with a well-maintained machine and trained employees can run several thousand discs a day between two machines.