you have sort of asked 2 questions there...
how do I back up Final Fantasy 10
answer -
http://www.psxcopyworld.co.uk/Tutorials.htm
use PrimoDVD or Nero, decent burner and decent media
without having to use a modchip?
As A03 said, knife trick. Jamming something into your PS2 tray and wedging over a clip and forcing the tray to open/close. Think of opening your front door or car door without the key. Can be done so many times before door is eventually knackered.