Memory card format

easy solution, don't buy cheep memory cards they have lots of reformatting problems. the cheep ones get damaged often and need to be reformatted, while the real ones will only need reformatting if you turn it off during a save, and then its only the one file is damaged not all of them.

Try take the card out, then put it back in fully. I've seen this problem before on an Official Nintendo GameCube 251 Block.
If there are saves on it, and it still asks you to format the card, just take the card out, and put it back in. You could also try a different slot...

Chances are, you just have to take the card out and put it in again for some odd reason. I've seen this fixing that issue.

I don't think any of the hardware, i.e. the GameCube or the Memory Card are damaged in any way.

Hope this helps.