Your temps are ok? you say you have tried unplugging every component. So I assume you have reseated your video card & ram and any other PCI cards you might have installed. How long has this problem been occurring and when did it start? I am not sure if this a new build or a system you had running for a while ok and now suddenly having this problem?
I would think that PSU would be enough, average gpu draw 150-230Watts I have 2x gpu that draw 190Watts each 3xHDD's running off a 750Watt psu, so 700 should be plenty for one card.
This could be several things, you maybe actually bluescreening but not seeing the message but normally you will get some pop up about it when it restarts saying windows recovered from some fault etc.
With a bsod your system normally restarts, if your system is simply powering off it suggests:
1. High Temp protection has kicked in and powered off the system after something has hit 90+ degrees. If there is a problem with a heatsink, heatsink paste or cooler the temp can ramp up extremely quickly (1-2seconds and it could be raging hot) quite possibly quicker they you will see in a temp monitoring problem if it powers off before you get to see the last second of two of temps.
2. Unstable overclock on CPU or GPU, put back to stock speeds if applicable and try again.
3. Faulty hardware, this could be anything from your psu, mobo, cpu, ram or gpu (more likely gpu given it happens when games are launched).
4. Insufficient power under load, so check you have plugged in the aux power connectors to your video card. Some have 1x6 pin some have 2x6pin make sure all are connected.
If the event of faulty hardware, this can be difficult to diagnose, the best way is try one piece at a time. Starting with the PSU since it is the easiest to replace and 50% of these problems can be attributed to a faulty or improperly connected PSU.
Ram you can try alternating the sticks of ram you have one at time to see if that makes any difference.
GPU even if you have an older model lying around from a previous upgrade it will do for testing purposes.
Lastly and least desirably the motherboard, only after you have tried everything else.
You just basically swapping one bit at a time until you find the culprit.
Would want to confirm if it is BSOD or simply powering off first though.