Short of taking off the back, I would contact samsung.

 

I Had this same problem. Couldn't turn on or charge. I had to take the back cover off and unplug and replug the battery. There is no other option when you try everything else. It turned on after I did this.

 

Don't despair yet, the same happened to me. Remove the sd card, then hold the power and volume up buttons for 20 seconds. Release and press power button again. That worked for me, at least.

Sorry, first post, cant post direct links

bigpicturebigsound.com/Samsung-Galaxy-Tab-3-Won-t-Charge-or-Turn-On---How-to-Fix.shtml

 

Yeah, that didn't work for me. I was able to get the back cover off and unplug and replug the battery. Works fine now

 

Great, I figured the video on how to remove the cover on that site would be handy if nothing else worked.

I lucked out, just taking out the sd card did the trick last time, but I saved the link for future reference because it seemed to have all the necessary info on one page