For what it's worth, this is a 3d version (finite volume, infinite surface are) of a phenomenon which might be more easily grasped in 2d (finite area, infinite perimeter).
And this idea of finite area described by y = f(x) [between this curve and the x-axis, say], with infinite perimeter, amounts to just the same thing some function f whose integral over an infinite range (e.g., from x = 1 to infinity) is finite. If you prefer to think discretely, this is essentially the same phenomenon as an infinite series with finite sum (splitting our curve into the block from x = 1 to 2, the block from x = 2 to 3, the block from x = 3 to 4, etc., the total area is the sum of the series of areas in these blocks, while the total perimeter is automatically infinite as at least length 1 on top and bottom is contributed in each of these blocks). So, consider, for example, y = 1/x^2, from x = 1 to infinity: infinite perimeter but finite area (as its antiderivative is -1/x + C, which only increases by a finite amount over this range; in the same way, differentiating any function with a finite asymptote yields examples, and every example comes in essentially this fashion).
In moving to 3d and making a surface of revolution out of our starting curve, we are now integrating πf^2 instead of f itself, so we need f^2 to yield a convergent integral rather than f itself, but otherwise everything is just the same. Thus, we can get away with f(x) = 1/x, as in Gabriel's Horn, but also, just as well, with f(x) = x^{-p} for any p > 1/2.
Not that these are the only such things; again, any convergent infinite integral or series yields examples. So, just as well, y = 1/2^x or such things would work.