My Sony Bravia suddenly went off and I rebooted it, but although it gets power, there's nothing on the screen.

If you wish to get some details; check the site linked here. Pull up older posts. Surf the site with patience. http://electronicshelponline.blogspot.com/
Faults & possible causes to LCD screen are given with illustrations. You can get an idea about the fault to an extent.

Somethings failed internally. Possibly a board or internal fuse. I assume there is no sound either?

you need to change the picture board/ the display board

Check and replace damaged component/s at it main power regulator [SMPS] section circuit. Contact any service technician. If you wish to get some details; check the site linked here. Viewing it in "Mosaic" or "Magazine" will make surf easy. Surf the site with patience. Pull up older posts. Do asearch by type in the brand name of your device at its search box.
http://electronicshelponline.blogspot.com/
It will be best to replace the power supply regulator board as card basis.

Unplug the set for a while, also a cable or sat box or home receiver and power back each to allow them to try to reboot.

If you get a blue screen at turn on and a short time later it shuts off--power supply is ok--set is shutting down as no signal detected to protect the screen.

If using a box first unplug it and wait and plug back--let box try to reboot itself.

Also as a test hook a DVD etc to a rear set of audio/video inputs and go to that input quickly when the set comes on (have DVD connected and on---see if set works that way.

No signal either a problem with the set or the signal source (box?).

$145 to come out and diagnose the problems it is a ripp off.The tv have solid power light to the tv.Useally 90% of the times is it the tv Tuner signal board is causes and the problems,that the board,where the incomming broadcasting signal cable plug into it.U have some basic machanicals skills like take things aparts and putting it back together again.$145 bucks u can buy a refurbish tv Tnuer signal to replace it ur self.Tries websites like Shopjimmy.com,Ebay.com to buy a whole refurbish tv Tuner signal board for the replacement.

U might need a new bulb.

same things happened to me...did you solve this?

COULD BE LVDS CABLE LOSE OR NOT MAKING A GOOD CONTACT , IF IS NOT THAT UNFORTUNATELY IT WILL BE NECESSARILY TO REPLACE T CON BOARD
YOUR RATING TO MY ANSWER ITS HIGHLY APPRECIATE
THANKS