I can't find any recommended specs for the linux version but I would guess not.
The Gamecube has a power yet slow processor and not a great deal of RAM, plus with GC Linux being a live disc it will take up more resources.
You could test it though, all it will cost you is a disc.
GC linux isn't installed on your GC but is loaded from the disc into RAM from which it then runs. It is partially possible to use an SD card as file storage (someone did that a while back to allow for some emulators and such I tink --> mame and etc.). Mostly one would use GC linux together with a BBA connected to a PC which then acts as the filesystem. It's rather slow though. But cudos to you trying!