UPDATE: Well, the PS3 manages to stay on for about 15-20 minutes before going back into dead mode.
I've contacted Sony and Console Doctor. Neither has replied. Console Doctor apparently charges £70 for YLOD repairs, and gives you back your old console. They pay for delivery back to you, but you have to pay for delivery there.
I've heard Sony charges £120 and gives you a refurb. That wouldn't be good.
Alternatively, am I overreacting, and is this a simple problem that I can fix?