Thank you all for your advice, I do so appreciate your help, but the problem most definitely is not dust build up , and the beeping noise is most definitely not fan whine.

The beeping noise is (or, I hope, was) similar to the sound made by a smoke alarm, to the extent that I didn't actually realise at first that it was coming from the computer, I thought it must be a neighbour's smoke alarm going off.

I did in fact have to have a new power supply fitted a couple of months back because the old one was overheating drastically amd causing videos to become unwatchable, but I always blow out the inside with compressed air at least every two months. The new power supply is fine, the case remains cool (previously it was really hot to the touch ).

It is of course ten years old, and has had its share of problems over the years, but just like with my cars, I will try to nurse them along until they give up the ghost.

Touch wood, there has been no evidence of the problem today, and it turns off like it is supposed to. Maybe it was a one-off glitch, and it won't return. If it does return, and Joe's suggestion does actually cure the problem again for a while, well, I can live with that.