I have had the same issue with my nexus 10 turning off and not powering on. The other problem I had was that the device would turn on but the screen would remain black and I would have to restart it. Here are the various steps I have found that worked for my device. They are in order of what worked quickest to longest process.
1-Hold the power button for 10 seconds. The screen should turn off and the device vibrate. Then wait 5 seconds and then push and hold the power button for 10 seconds and the device should vibrate and screen come on.
2. Do the same as step one but hold the power button ranging anywhere from 30 seconds to 1 minute. The device should power down and then you can wait 5-10 seconds and then hold the power button and start the device as normal.
3. When the above didn't work I noticed that if I plugged the tablet into my computer, then it would start using the above methods. I have read that it works using the power outlet as well, but that has not worked for me.
4. When none of the above worked, the last step Google had me try was to hold the power button + Volume (both up and down) at the same time. I held this for one minute. Then the device powered down. Continue to hold and a screen should come on. On top will be two arrows, on the side a green start button, to the right of the green button some device info, and in the center of the screen will be the droid icon. Try double tapping the start icon. If that does not work. Push and hold the power button for 10-30 seconds. The device should power off. Wait 5-10 seconds then power on and start your device. Hopefully it will start as normal.
These are all the little quirky things I have read in forums and have had to use to keep my device turning on. I have heard that this is a software issue and that a fix is being worked on to prevent this issue. As to the truth of this, I am unsure.