Well, there a couple things that could cause this.
1- Your memory card could be the problem. It could be anything from it having gone bad, to it simply being dirty. So, take a Q-tip or toothbrush, put some acetone on it, and give it a little rubdown. Let it dry before you try it again. If it works, awsome. If it dosent, try another memory card and see if the problem continues
2- If the problem continued with the 2nd memory card, then something is wrong with your game cube. Either the greenboard that reads the memory cards is going bad, or it is something worse, affecting the motherboard. If this is whats happening, then you only have two answers. Either send your GC to Nintendo for repair, or go buy a new one.
Thats all I have for you...