A few ideas and things to try:
1. Unplug the PC from the wall, leave the PSU in the "On" position. Now press the power button on the PC itself 10-times, slowly. Press it down, count one second and then repeat. After that, turn off the PSU, plug the computer back into the wall, turn on the PSU and see if the computer powers up.
2. Unplug everything from the PC, pick it up and give it a shake or a turn, whatever, just move it around. Hear any rattling? It's possible something got loose and is causing a short.
3. Move the PC to another room, plug it into the wall and see if it works there so we can rule out a bad ground or something in your wall plug.
Did you build this machine yourself? If so, what is the PSU in the machine? It's always possible that it had a short and failed, especially if it's a cheaper brand or one of the "lesser" PSU's from a known brand.
Either way, two things to try:
1. Remove the motherboard/PSU from the computer and set it on a static free surface, such as a wooden table. Connect the power supply to the motherboard in the appropriate spots (24-Pin and CPU), plug in the PSU and find where the power switch plugs into the motherboard, it'll be two pins on the board itself, your manual can tell you where. With a flathead screwdriver, touch the two power pins with either side of the flat part, which will jump them and cause the machine to power on (assuming your PSU hasn't failed). If it powers on, we know there's a short somewhere in the case, so your next steps are to figure that out. Reinstall your motherboard and see if it powers on.
2. Your PSU may have failed. Do you have a spare? If not, buy one (figure out the return policy, in case this isn't the cause of your issue) and install it to see if the PC powers up.