OK. So if you feel OK with following instructions to go inside the case, go through a few suggestions:
Just unplug, press the power button (it won't start of course, we just want to accelerate the draining of power off the board) After a couple of minutes try to restart.
If no good, unplug, open side cover and remove graphics card. There are probably screws holding it on the end and the connector may have a plastic latch also. Try a boot from integrated graphics.
If still no good, unplug again and remove the memory sticks, then re-insert them. Try again.
Then you can try a reset of the CMOS by locating a coin cell battery, and removing it by gently pressing it's retention clip. Leave the PC unplugged and let the power drain for a few minutes. If it does start you'll have to reset the time and date.
IF there are multipe memory sticks, try one at a time
Past this, you'll need other parts to test with (a different power supply, etc). It's a somewhat older system by the looks of it, hopefully the board isn't failing