Intel P4s were the cpus that ran hot for the most part. But in addition to a cpu overheating and the protection circuit kicking in to see the system shut down you could be looking a power supply or board problem if the cpu isn't taking a dive on you. A constant beeping does point at a cpu not holding up any longer under a good load.
The first thing to look at is the hardware monitor in the bios to see just what is being seen there for a temp. Is this a Socket A XP2400+ you are referring to there? If the cpu has a stock heat sink/fan combination on it the thermal pad is probably letting go from age where a 3rd party cpu cooler and a fresh application of Artic Silver 5 or MX2 on a Zalman 7000b model cooler will see temps brought right down.
The other thing that will often see a system shutdown when placing a good load for power doesn't make any beeping sound namely the power supply not seeing good regulation. For XP and Vista alike one program called PC Wizard 2008 can display cpu, board, and even the video card's temps while in Windows and usually is right on with the hardware monitor. That's a free program found at CPUID