what size supercapacitor do i need to replace 4 3.7v 2400mah batteries?
If you're trying to power a circuit off of capacitors instead of a battery for a long while it will not work. The two devices have different physics in terms of their energy storage. A capacitor's voltage does not change instantaneously, but it can supply a large amount of current instantaneously and discharge rapidly.

What are you trying to power, and for how long?
-Ryan

Here you go, just as Adam said, this will be a very large capacitor. My solution only brings you to a half charge because I assumed 500mA.