Player-Owned Shop in Animal Crossing (GameCube): Not Possible—Debuted in New Leaf

Not possible. The original Animal Crossing (GameCube, 2002) only lets you buy from Tom Nook, Able Sisters, and Tom Nook's catalog. You cannot open or manage your own store.

The feature debuted in Animal Crossing: New Leaf (3DS, 2012) with Re-Tail—a consignment shop you customize and run.

On GameCube, your "business" options are limited to selling fossils/fish/bugs to Nook and designing patterns at Able Sisters.