Mac Mini not POSTing, Fan spinning at full RPM.

Wow you are a hold out with your Elsel don't get me wrong at its time it was the Cats meow! It was the very first Intel MacMini!

Both hardware and software has moved on in leaps and bounds since then! Other than a fun relic running old software or as a found remembrance. There are better versions of the MacMini available migrating onward to holding on to what you have currently on the drive software wise. While not easy to find I think you would find using your older stuff you have on it just better. Some of the older Gaming gets better performance (CoreDuo) if that's your kick.

From your description SMC is not happy as one or more of its sensors on the logic board are out of bounds of what it expects. As you are not getting any video response is likely the combo CPU/GPU this series used gave up. It wasn't uncommon for the GPU logic to fail. Even if the logic board has a deeper issue it will require deep skills to diagnose and that can get expensive!

So... Replacing the CPU would be your first option locating a still good chip this system can support. Sadly, not likely as they are long since gone.

The other option is locating a newer (used) MacMini which supports the highest version of OS-X this system ran (Mac OS X 10.6 "Snow Leopard") unless you are still running OS-9. Then you have a painless transition just swapping the drive over. That's what I would recommend here..

And while still even more costly jumping to one of the newer M Series (used) MacMini's. But that will require running to a much newer OS and Apps as well.