A quote: "Elementary calculations, with calculus, allow one to show that although the Horn has finite volume, it has an infinite surface area. This is considered paradoxical, because it says that although an infinite amount of paint is required to cover the interior surface of the horn, the entire interior can be filled up with a finite amount of paint." [Emphasis added]

It would have been much clearer to have said, "The volume of the horn can be filled with a finite volume of paint." There's no reason to use language seemingly designed to create a aura of mystery.

When I was an undergraduate at Hampshire College my advisor, David Kelly, used a draft version of the linked text in an experiment to teach introductory calculus using nonstandard analysis rather than limits. This was probably in 1978, and I was a teaching assistant for the class. I think the experiment worked pretty well.

The snake example is good, but for me it is more intuitive to think of the z-y plane as having very small volume but infinite surface area as the plane extends infinately in both z and y.