TravelMate 5520 overheats, CPU fan spins then laptop powers off, Is it worth repairing?

I wouldn't be terribly tempted to spend any money in trying to figure it out. At close to 20 years old it's not going to end up being anything very useful. If you do want to dive into diagnosis though we can look a little at what actually happens when a laptop is first turned on.

When you hit the power button, the motherboard tells either the battery or the AC input to provide power to the rest of the motherboard. That often can be seen by the CPU fan turning on, as well as some status LEDs. When that power is applied to the motherboard it powers up the CPU and memory. When the CPU initializes it then tests the first small portion of memory and copies the contents of the BIOS into that memory and does a couple of housekeeping tasks, such as turning the fan back off or setting it to a slow speed.. All that happens before the first code is actually executed, it's instead built into the CPU and chipset. Once the BIOS code is copied to memory the first actual code execution happens and POST (Power On Self Test) is started. The first thing that POST does is initialize the rest of memory and run a quick test on it to verify things are working as expected. If that test fails, you typically get a continuous beep sound.

On your system it sounds as if the CPU is initializing, which turns the fan off, but POST isn't getting started. That could be a corrupted BIOS or a failure to load the BIOS code to that first portion of memory, or an issue with the first portion of memory not being able to initialize. I'd guess the former more often than the latter. Try replacing the CMOS battery, if you haven't already, they will typically last for 7 years or so, and you could be ready for your second or third one at this point. Then do a CMOS reset. There is likely either a reset header and jumper on the motherboard, or a pair of reset pads. You can sometimes also do a reset by shorting the leads of the CMOS battery for a minute or so with the battery removed. Once it's been reset, put it back together and try again to turn it on.