Flashing red light when turned on, powers off by itself

Unfortunately, these old consoles have a bad reputation of the capacitors on the motherboard and power supply failing. When they fail, they leak their electrolytic goop onto the circuitry below them causing corrosion, and eventually can dissolve the traces. Notice how all of the components around C7G10 look pretty gross but further ones (like C7G8) look cleaner? That's because the closer ones got soaked in capacitor goop. Yours looks repairable from the photos, but will probably need a re-cap and a good cleaning with isopropyl.

This is a great opportunity to learn / work on your soldering skills if you want to invest in a new set of capacitors, but not the easiest job for a complete beginner. There are a lot of guides for re-capping these original Xbox's, but consider the possibility that the capacitors aren't the only issue.