Although I suspect that the hard drive may be another of those dying Quantums, or at least one with corrupt system software, judging from this comment:
Is it true that, without a floppy inserted, the machine makes a sound like something stopping (there's a noise which gradually drops in pitch until it's gone completely and all you can hear is the fan)? In that case I'd copy everything I can off of the hard drive and make backups of the machine's system software and any installed applications before the hard drive dies completely.