This sounds like it could be the logic board. If it is, and you have access to another Mac with FireWire, you may be able to get the iBook to remain responsive in
FireWire Target Disk Mode long enough so you can retrieve your data by this method:
If the iBook is on, shut it down and wait 30 seconds (or longer). Pick it up in your left palm to the left of the trackpad and squeeze the case there between your palm and left thumb as you push the power button and key combination to start up in FireWire Target Disk Mode. Do not let up the pressure on the case. Continue applying this pressure as you save your data to the other Mac. If you let up and the iBook dies or freezes, you will have to start over, allowing it to rest several hours or overnight before beginning again.
If you determine that it is the logic board, you can follow the instructions elsewhere in this thread to explore repair options under the
iBook Logic Board Repair Extension Program.
Let us know how things go.