G4 powerbook won't charge

Try removing the main battery and leaving the G4 on the mains for a couple of days. The problem is likely the internal backup battery. The internal is (well, was when new) rechargeable and, if you pull the main battery, all the current goes to the backup. It is possible (but no sure thing) that having the main one removed will allow a bit of charge to bump up the backup.

Many PBs now have dead or weak backup batteries. New ones are only available for the latest models and then only for the 17-inch version. People sell used ones but, to me, buying a used battery is like buying used chewing gum!

To add insult to injury, replacing one requires a partial teardown that is beyond the average user's tolerance level or skill set.

If you wish to keep using this computer you will probably have to buy a new main battery. I have a 17-inch 1Ghz PowerBook from 2003 and it has a relatively new Apple battery I bought shortly before Apple discontinued that battery. Even with a known-good main battery, the dead backup battery will not allow the G4 to start with a fully discharged main battery. I have to wait a few minutes until a little current gets to the main one before it starts

The only place I would trust for a new battery is Other World Computing (macsales.com) because they have a solid return policy and excellent customer support. We don't know which PBG4 variant you have but this page has their listings for G4 Powerbooks if you scroll down a bit:

http://eshop.macsales.com/shop/Apple/Laptop/Batteries

Note that the cost is not trivial, so I think you will be weighing the cost of a new main battery that may not fix the issue completely against the cost of a newer computer.