There are several things that can cause this.
The most common is the horizontal circuit, when it shorts out it will shut the set down just like yours is doing. This repair normally runs $200-$300 depending on what is found.
However this could also be caused by the vertical circuit, power supply, and high voltage circuit to name a few. The average repair price for any of these is $125-$300 depending on exactly what is found.
However to determine exactly what the problem is will require troubleshooting. Some knowledge of electronics and a multimeter will be needed for this. If you have this knowledge/tools I will be more than happy to walk you through the troubleshooting of this set and determine what is wrong. Also most repairs will require good soldering skills so you will need these as well for us to repair the tv after diagnosis. If you don't have any of this then a repairman would be your next step.
I am sorry to bring you the bad news on this one, you definitely have an internal failure in the set that is putting it into shut down.
If you would like any additional information or have more questions please don't hesitate to ask!
Please never accept an answer until you are 100% satisfied and feel you can leave positive feedback. If I have answered your questions and met your needs then please press the green "accept" button. Otherwise please feel free to ask anything else to get your problem resolved. Positive feedback is always appreciated! Thank you,Tim Nolan