Unwind a paperclip until all you have is a straight piece of wire.
Take a set of pliers and bend one end of the paper clip 1/4 inch inward. You've now got a makeshift key.
Stick your paperclip "key" in the lock of the makeup box and wiggle it around. Your goal is to find the latch within the lock and press on it sufficiently to unlock it.
Continue to slowly move the paperclip around, tapping it against the insides of the lock. Be patient. You'll know you've hit the latch when the lock clicks open.