Q: why won't apple tv turn on?

If you unplug it for a while then plug it in, does the light flash then go solid?

No light activity (black) means the unit is not powering up.  Sounds like it is dead.

Are you sure you are plugging the unit into a powered outlet and not one switched off?

Are you sure you have pushed the power cable fully into the back of the unit?  It is a tight fit.

With no light activity at all, it sounds like it is broken.  Probably a bad power supply.

As a last resort before throwing it out, you might want to try a software restore, but with no power, it will not work.

Read this article and as is suggested, take it to an Apple store and have them evaluate it and attempt a repair.

Restore your Apple TV through iTunes - Apple Support