Erase HD when both Disk Utility and Terminal fail.

You may be able to open the CD tray on startup:

Eject CDs and DVDs from your Mac - Apple Support

Of course, then if the hdd is failing/failed you still won't be able to erase it.

There are tear down vids available for that model. It doesn't look difficult at all. If the plan is to dispose/recycle the computer, it might be an interesting project to disassemble it and remove the drive. Then you can simply destroy it.

Google "iMac G4 teardown"

It may well not start because the hard disk failed, and which would also account for the other symptoms reported.

Scrounge a replacement hard disk, swap it, restore the backup, and hand it back to the owner. 😈

I can't open the CD drive (and so use an OS Install Disk).

It's not clear to me whether that means you are unable to open the tray or the media in the optical drive does not mount.

Holding an Option key on boot as D.I. Johnson suggested and described in Eject CDs and DVDs from your Mac - Apple Support is a good idea but it will only work if bootable media already exists in the tray. Instead, connect a wired (USB) mouse to the iMac, then start the Mac while pressing and holding that mouse button.

That action will cause the tray to open on startup.

• On slot loading Macs, it causes the eject mechanism to cycle, ejecting anything in the drive.

• That feature was included to force eject unreadable / unmountable media.

While I'm on the subject the Terminal command

drutil eject

... does the same thing. Of course the Mac has to be running first.

Thanks for catching my slip, there, @John Galt. 👍🏽

I'd forgotten the little trick w the mouse.

Not a slip, it's just long forgotten computing lore. Apple no longer mentions it anywhere. Like optical discs.