My friend's PS2 was doing this.. Luckily I was able to fix it..
It's simple, there's four rubber feet on the bottom of the PS2. Pull them out, along with the two little tiny plastic squares that are the same size.
Unscrew the screws inside and take the cover off CAREFULLY!!!
To take it off, make sure there's no memory cards or controllers in, then pull the back of the cover up a little bit and pull off the LEFT side of the PS2. There will be a little silver strip attached from the case to the inside, rip off the little sticker, but be INSANELY careful not to break the silver strip! This is what turns the PS2 on and off.
Now.. get an eye-glass screw-driver and take off the disc cover, then open the disc tray by pressing the blue button. Shut off the PS2 from the back so the tray stays out.
There should be a little white dial. Move this a couple clicks in any direction and try running a game. It's safe to run a game with the cover unscrewed, but make sure it's laying on top otherwise the disc will spin out.
Keep adjusting the dial unti a game works, then put your PS2 back together.