When trying to start, I get a "honk" instead of normal chime and then the light flashes occasionally from dim to bright but computer does not turn on

Here is an old Mac trick that sometimes works. Turn computer on. Hold down (at the same time) Control (Apple key) Option P and R keys. Yes, you may need more than two hands. Hold those keys until you here three or four chimes. Release and wait for the computer to start.
Another trick: Hold down the Option key when starting. You should see your hard drive logo on the screen. Select it, then the arrow on the right and see if it starts.
If you have a backup firewire drive (and you should) plug in that drive, start with the Option key pressed, and see if both drives show up as icons. Select the backup drive, Not plug and play, more like plug and pray,
If none of these tricks work, you could have a memory issue, a power supply issue or a motherboard issue. G4s are real cheap now. $40-$60 You can always pull the hard drive and mount it in another computer. And for your own protection, back up that hard drive to a bootable external firewire drive.