My iphone 4 has been working great, just went to make a call and its dead. Won't turn on?

I don't think you can remove the battery, if you can remove it and keep it out for 1 minute

I am guessing that your phone probably shut itself down to prevent data loss. When was the last time you shut it off by holding the top power button until it says slide to power off and then powering it off? I may be wrong in this but,... Okay, so have the phone unplugged and I want you to press and hold both buttons at the same time, the Top Power and the Home Screen Button and just continue to hold them, sometimes even for like a minute but I'm guessing within like 30 seconds or so the Apple logo will appear then you can release the buttons. If this does work you can prevent this from happening again by power cycling your phone on a daily basis or at least every other day. :-) I hope it helps good luck.