This is most likely an issue related to power. This specific behavior is caused by the console failing to perform pre-boot standby checks. Some necessary standby voltages aren't being created, so it turns itself off.
The first thing I would do is unplug the hard drive. If it works, your issue is likely the hard drive and replacing it should fix it. If it still doesn't work, you can try it with a new power supply.
If it STILL isn't working after replacing the supply, it's more than likely there is a board-level failure. The most common culprit I see for this problem is this GAS chip on the motherboard, and much of its related circuitry.
I don't have any schematics for this console, but I believe this particular chip is responsible for producing loads of necessary standby power rails from the 12v coming out of the power supply.
Most of the time I see this problem, this chip is damaged due to liquid. It resides underneath the fan, and is in a location where liquid ingress can easily reach it through the fan grilles located on the exterior of the console. Sometimes, cleaning it up and giving it a reflow is all it needs. Other times, the chip itself needs to be swapped with an identical replacement.
For example, here is the last one I had. Liquid had clearly managed to get right on top of the chip. Some kind of juice or soda, as it was VERY difficult to clean even with flux and reflow. However, that ended up being all I had to do. It booted right up after cleaning this area and touching up the chip leads with an iron and flux.
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