That could be as simple as a hard drive going bad...only other thing that can make a grinding noise is a fan.
I would go to OWC, http://www.macsales.com and see about a new hard drive for it, and also an external enclosure. Attach, format the new hard drive in the external enclosure, use something like Carbon Copy Cloner or SuperDuper to make a bootable clone of the internal drive on the new drive.
Shutdown and remove the internal drive and replace with the new drive, place the internal drive in the enclosure so you have a backup, even if it is failing since you won't use it often but why throw it away if it hasn't failed completely
Since the older iMac G4 PPC hardware would be able to boot from an external FireWire enclosed hard disk drive, check out those models of enclosure with the oxford-type chipset and also includes a power supply so it won't be attempting to load an OS X from an unpowered HDD. The external often has to be turned on before the computer, so a bus-powered external drive or a USB drive will not work to run an external system, be it a clone or a complete installation from DVD via the computer.
Note the internal HDD is a PATA or ATA/IDE, as finding these is more difficult now. Also there is a legacy OWC model SSD for older model upgrade, but this may be a stretch. If you get an external enclosure, it may be for an SATA type HDD, so this may not be swapped into the iMac G4, since it uses the older PATA or ATA/IDE hardware... a 3.5" drive of up to 7200 RPM could work, if not too warm.
There also are issues in these models that my include hardware beyond the hard disk drive. That is a place to start, and if you go into there, be wary of a need to do an involved take-apart, thermal paste renewal, and major attention to detail in order to complete the mission successfully. Also, there is a battery to replace on the logic board. The 3.6V 1/2AA Lithium battery powers the PRAM and system clock, and may affect the ability of these computers to start up when the battery has finally failed. They are good for a few years, but may limp on if the computer remains plugged into the mains. After a power-interrupt is when this battery (and reset) has been known to be discovered to have failed.
When replacing the hard disk drive, PRAM battery, thermal paste; in that vicinity is where one may access the upper RAM slot and consider in the 1.25GHz iMac G4, it can use a 1GB chip of correct spec standard RAM; so the total in these can be 2GB with the lower slot using a laptop upgrade chip of 1GB. PC2700, to Apple spec, to be sure there are no odd issues due to poorly made or incorrect spec cheap PC RAM.
•iMac G4 - mr totes:
•iMac G4 Take-apart for Drive and RAM upgrades:
•FW self-powered HD enclosure with oxford 934 chip from OWC:
•Memory for iMac G4 1.0 & 1.25GHz models
•3.6V 1/2AA Lithium battery:
Hopefully this information and sample references may be of use.
The easiest thing would be to run off of an external firewire drive.
You may want to run these "standard" fixes if the problem persists.
1) Check the amount of free space on your harddrive. You should have a several gigs free.
2) You should run disk utility
Macintosh-HD -> Applications -> Utilities -> Disk Utility
a) verify the disk
b) update your permissions.
3) Try a safe boot.
Shutdown your machine. Hold down the shift key. Poweron. Wait awhile; wait awhile while you harddrive
is being checked.
4) Another way to correct filesystem problems is into single use mode.
This page will tell you how to get into single user mode.
Basically, you hold down the command + s key then powering on your machine. The command key
has a little apple symbol on the lower left. It is between the alt/option key and the space bar.