If it just fails to turn on, the main power supply is probably at fault and without some technical background and a couple of test instruments, you will not be able to do the repair yourself. 
What you can do though is; unplug the set, let it stand overnight or at least several hours, remove the back panel and look for a separate board with mostly larger components on it; it will have a ferrite core transformer on it and several larger capacitors which are cylindrical and mounted vertically to the board.
 
Before touching anything inside, especially on that board, allow several hours to be safe since the larger capacitors can hold a painful voltage charge for some hours.  
 
If you see signs of overheating such as discoloration of the board material, it may not be repairable but if you can find a part number on the board, you might be able to buy the power supply and install it yourself.
If there are no signs of heat damage to the board, there may be companies offering to repair it as a subassembly instead of repairing the whole set which could be much more expensive.
If you choose to replace or have it done, the power supply still may not be at fault since most are capable of switching on and off if a downstream problem is causing an overload.
The clicking you hear is a relay that repeatedly connects, then disconnects the AC to prevent fire or further damage.