It sounds like your power supply has failed. This could be caused by bad capacitors in the power supply itself, or it could be caused by some other circuit board, likely the Y-sustain or Z-sustain board, or both.
This solution assumes you feel comfortable working on the tv yourself - if not, it will need to be fixed by a service technician.
Begin by removing the back cover. You'll need to take out all the screws that have an arrow pointing at them. You'll only need to remove the large cover, not the smaller one where the audio/video connections are located. Put the screws in a plastic sandwich bag so they don't get lost.
The power supply board is in the center top area. Check the tops of all the capacitors. They should all be flat. If any of them are bowed out or deformed then you have a bad power supply board that will need to be replaced, or replace the caps if you're capable. If the caps look ok then check the fuse at the bottom right of the power supply board. Go to the appropriate paragraph below.
1) IF THE FUSE IS GOOD:
you likely have a problem with one of the sustain boards. Unplug the connectors on the Y Sustain (Ysus) that sit between the two fuses. Plug in the tv and see if the power supply board comes up. If it does you'll hear a click and see the green light on the front begin flashing. This indicates one or both of the sustain boards has failed.
To determine which board has failed, reconnect the Ysus connector from the power supply and disconnect the connector at the top left of the Zsus board. Try turning on the tv. If the power supply comes up then you have a problem with the Zsus board. If it doesn't come up you have a problem with the Ysus board and/or the Zsus board. The power for the Zsus board is routed through the Ysus board, so if the Ysus board is bad you won't be able to check the Zsus board.
Many blogs recommend replacing both if one is bad because they're both high failure items, and replacing one without the other means you may have to go back and replace the other after a short while.
2) IF THE FUSE IS BAD:
You may have a problem with the power supply board. Check the caps for any deformed tops. If so, you'll have to replace the board or the caps, depending on your skill level. After making the replacement, disconnect the power to the Ysus and verify you have a good power supply board.