It sounds like your battery is at the end of its life. I have seen a lot of issues with Samsungs batteries not lasting as long as expected. Some have even gotten a replacement battery for free from Samsung. Not saying everyone is that lucky though.
Some times you can tell by looking at the battery and seeing if it looks like its puffed up a bit in its center. You can also take the battery out and try and spin it on a table. If it spins like a top(easily) its probably not a good battery. This is a sign of over charging a lithium ion battery, or not using the correct charger for the specific battery in your phone.
You can try taking the battery out and then putting it back in after a minute or so, and try and power it up.
If you know anyone with the same model of phone as yours you could test their battery in your phone to see if it powers it up, then you would know for sure it the battery. If you can't find one to test I would suggest getting a new battery.