I want to design a battery that holds at least 20 kWh can charge something in 15-30 minutes but I want it to be as small as possible. What is the smallest size it could be built at and how much would it cost?

Look up, read and understand the definition of kwh. It is a rate not a quantity.

You will need more skills than this guy. Stanford researchers seek 'Holy Grail' in battery design (http://news.stanford.edu/news/2014/july/pure-lithium-battery-072914.html) Yes and a few million dollars for research. This is not a simple task.