That's not really the paradox. The amount of paint required depends on the thickness of the coat, and the inside is forced to get a thinner and thinner coat (since the space available gets thinner) while the outside is assumed to be painted with an even thickness of paint. If you painted the outside with a layer of paint whose thickness is proportional to the thickness of the curve, you could do it with a finite amount. (This is all assuming idea "paint" that is continuous and arbitrarily subdivisible, real paint is made of molecules.)

The intended "paradox" is that the surface area is infinite but the volume is finite.