Chosen Solution
A few months ago my xbox would shut down after about 40 minutes of gameplay and the light in the center would turn red making the only way to turn it off completely be by plugging it off the wall. Now when I try turning it on, the light on the power brick goes from orange to green for a split second and the xbox’s light from green to red as well. The fans on both the xbox and the power brick spin for as log as the lights turn green but then fail and stop spinning.
Your X-Box is overheating and it’s shutting down to prevent a fire-hazard. You should disassemble the device and clean the fans out, as well as perform a Red Ring of Death fix using this toolkit: Xbox 360 Red Ring of Death Fix Kit The repair isn’t guaranteed, but it is your best bet at getting a working device again.