ELI5: How can Gabriel's horn have a finite volume but infinite surface area?

Gabriel's horn can hold a limited amount of paint inside it, but you would never have enough paint to coat the outside of it because the horn gets endlessly narrower and longer.

Imagine you have a single ball of playdough:

• Squishing the dough: If you roll that playdough on a table, it becomes a long snake. As it gets longer, it gets much skinnier. No matter how long you stretch it, you still have the exact same amount of dough you started with. This is like finite volume—the amount of space inside the horn is trapped and never grows past a certain limit.
• Covering the dough: Now, imagine trying to cover the outside of that playdough snake with tiny stickers. When the dough is a short, round ball, you only need a few stickers. But as you stretch that same ball of dough into a super-thin, miles-long thread, you need millions of stickers to cover the outside.

• The space inside shrinks away to almost nothing very quickly, so the total amount of air or liquid it holds tops out at a fixed, finite amount (about three cups).
• However, because the horn stretches on forever without ending, its skin continues forever, creating an infinite surface area.

In everyday life, physical things like paint molecules have a real thickness, so you cannot make a real Gabriel's horn. But in the world of pure shapes, you can easily have a container that is small enough to fill with paint, yet too big to ever paint the outside.