I have a very old Mac OS X 10.4.11, the only use I have for it is that it is attached to a CanoScan 9000F Scanner. It has the side entry for the DVD disk, which no longer works. Attached an exteral drive, will play but will not burn a disk.
Is the external drive a DVD burner?
There were early drives that were readers (only), or that could burn CDs but not DVDs. There were also different flavors (“+”, “-“) of drives and media.
That vintage Mac wants to retire.
You should let it.
Erase it and recycle it.
It's very possible that the external optical drive is not compatible with a 20 year old OS and vintage Mac.
I have a very old Mac OS X 10.4.11,
We do not know from "a very old Mac OS X 10.4.11," what exact iMac sub-model you have, but many were fitted with optical drives having a "fail-safe" device. If the drive fails, an internal block would deploy to prevent insertion of more disks. If you cannot insert a disk at all, the drive is dead and not recoverable. If it accepts a disk and then spits it out, it could be simply dirty.
Slot-load optical drives get filthy, especially if seldom used. Before giving up I would invest US$5-10 in a drive-cleaning disk and see if that wakes it up.
I have used a a cleaning disk to successfully resurrect seemingly dead slot-load drives in older Macs. In three of the cases an Apple tech had examined the computers and pronounced the drives dead and in need of replacement at a cost, at the time, of US$140.
Worth a try.
Mac OS X 10.4.11 DVD drives not working
