In the absence of Time Machine, you have a number of options but all of them involve using media, which is not the internal drive, to store the backup.
Your choices are an external hard drive connected via Firewire or USB, Optical Drive (CD/DVD) or Tape Drive.
If you plan to run regular backups as opposed to just the occasional drag and drop method, you will also need Backup Software. Google for that.
I note that you run 10.5.8 which means that you should have Time Machine installed on that G4. This is by far the easiest solution for you.
You will need a large capacity external hard drive, either Firewire or USB. Both Western Digital and Seagate make large capacity, low(ish) cost, drives. 1TB should be enough for you. It does not have to be a Mac formatted drive, Time Machine will require the drive to be formatted as Mac and will take care of that.
Just plug the drive in, configure Time Machine and off you go. Automatic backups from then on.
P