Why does a 12 inch Macbook refuse to turn on with a lot of battery remaining unless it is plugged in?
Assuming it is a 12" MacBook (implying at most it's still a little shy of being a couple of years old) and not a 13" model (which could be a lot older), then the first thing I'd do is reset the System Management Controller.

For a 12" MacBook you reset the SMC as follows:


Shut down your computer 
Keep the power cable plugged in. 
Press at the same time shiftoptioncontrol 
(on the left side of the keyboard) and the power button 
Let go 
Turn your computer back on with the power button. 


Once you've done this report back on how it went.

If in fact it's not a 12" MacBook, then please update your question with the exact model so we can make sure we're providing accurate advice for your model.