I don't know if this helps. My CD player had the same problem and it would eject by itself. The problem was with the "arm" that's suppose to lift the CD and hold it in place. It was stuck, and if the CD isn't lifted up, the player would eject automatically.
So my guess, open up your PS2, see if the CD lifts (the older systems had a white plastic/opening where the "arm" goes up. If it doesn't lift up then you know there's a problem with that part of the PS2 and you can investigate further.