Try shutting off the system by holding the power button down for about 10 sec. Once off, unplug the power from the back of the system and press the power button. You might see the fans jump as it consumes the residual energy available. Plug it back in and try to boot it.
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Edit (Additional Steps to try):
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Do you have another, know to be good, power supply you can try in the system?
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Since it turns off after 10 seconds, that wouldn't be long enough to attempt OS boot. This sounds like a hardware issue.
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Reasons for a quick power off could include:
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* Bad power supply cannot sustain required voltages
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* Short-circuit or other fault detected by motherboard power circuits
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* CPU or GPU device is seen as overheating.
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Also, you might try re-seating and PCI/PCIe cards, power and data cables, etc within the system. Clean out any dust you find as you go.
Try shutting off the system by holding the power button down for about 10 sec. Once off, unplug the power from the back of the system and press the power button. You might see the fans jump as it consumes the residual energy available. Plug it back in and try to boot it.