Similar thing happened with my 4. The phone just randomly died one day.
The phone came back to life after a week. I just left my phone alone for a week, and then plugged it to the charger for 24 hours or so. And one morning, I woke up to a vibrating sound of my 4. Ironically, I was scheduled to buy a 5S that same morning.... In the end I just bought a new phone.
Here are some things to try:
First try connecting to a computer, and see if it turns it back on.
And Try hard reset (push home button AND lock button together for 30 seconds or so). Do it multiple times.
Leave the phone out in the cold weather for a couple of hours. Then bring it back inside, and let it warm for several hours. Then charge the phone, and leave it plugged in for 24 hours or so.
Or leave the phone alone for a week (if you can live without a phone for a week, or if you have another phone), and charge it afterwards.
Buy a new phone.