My only guess is that you were sold a cr@ppy PS2 and the memory card slot/reader doesn't work.
I had a similar experience only it was the controller ports. Some metal sticks were bent out of shape. They just needed straightening out.
But this may be different.
First, I suggest you try the official Sony 8MB Memory Card.
You could also try the other memory card slot.
Repair might not do much good though--Unless these people can give a new memory card reader.
Here's an experiemnt though. With the PS3 ('Cos that doesn't have memory cards), there is a PS2 Mem Card to USB adapter. You could plug that in the USB port (It's located to the left underneath the controller ports, near the bottom) and see if it works (Unless your USB ports are cr@p too)
Hope this was helpful
BBBE99