Can't turn on my iPhone 4 . Tried pushing both buttons. Help

You do not simply push both buttons, you need to press, at the same time,both the power and home buttons and hold both buttons until the Apple logo appears, about 10-40 seconds, then release both buttons.

if that does not get the iPhone to start, take it to an Apple Store Genius Bar so the techs can test it.

Also worth asking, is it charged? If the battery went flat, it will not start, period. Plug it in with known-good Apple or Apple-certified cable and charger for at least 30 minutes to make sure there's juice inside.