I installed Mint 11 LXDE to an external USB, but this laptop doesn't boot from
USB in the BIOS either. So I used a Hiren's Boot CD, since it has the PLoP Linux
Menu on it. It has digital media slots, but I don't know if they'd boot. You may
want to try something like this, too.

Grub (legacy at least) came with an image that would boot from floppy.
I'd be surprised if the lilo bootloader didn't have a floppy boot-image
too.

Of course there is always puppy Linux which boots from cd and loads
entirely into ram. It should be possible to keep the save-image on a
thumbnail drive.

What you're talking about is a chain-loader but if the idea is to be
able to use the USB device then be able to install software and use
this PC then your better option would be to fit a hard disk IMHO.
Presumably it'll only have IDE ports but these drives are still around
even though SATA is the more prevalent now ( and SSD's the 'New Kid on
the Block' ). Even second hand drives abound if funds are an issue,
even if it's not being worth spending much on an old system.

Yes it is a small part of the issue. This PC has been used for
my ham radio efforts after it was given to me with a bad hard
drive. I did have an extra which I used loaded with 10.04, then
11.04 then 11.10. It then failed and I found another old drive
I used again until it has failed. They do wear out I guess. I have
tried a couple of the local computer repair shops to see if I could
find a 30 or 40 GB drive to no avail yet. I may end up trying one
of the on line shops to see if I can get something to add back in
to the box. Then I got to thinking about a work around. I suppose
I could maybe use a dedicated ham live CD, but I thought that I
could get a USB drive of about 32GB which would be a little faster
than the live CD but not as fast and the hard drive, and it would be
writable too.
And it allows me to keep learning and thinking.

You could add a SATA or SCSI controller, perhaps it'll be easier to find one of
those hard drives from an old server motherboard. Remember the old Mac G4 towers
used IDE, too.