Sounds like a power problem. Have you made sure that it is charged up? Try using a different cord to charge it and see if it works. If that doesn't do it, the battery may need replacing.
If neither of these works, you could try a "Factory Reset", which will erase some things, but not all. Located the pinhole on the phone and use something small (but not sharp, don't want to damage it) to depress the button and then try starting the phone (if it doesn't start on it's own).