Non-mobile: https://en.wikipedia.org/wiki/Menger_sponge

You can do the first few levels with business cards:

http://theiff.org/oexhibits/menger02.html

As you go to higher levels, that "infinite surface area" thing starts to become difficult to manage. ;-)

Or equivalently any object with a fractal dimension between 2 and 3.

I think Sierpinski's triangle may also be of interest here, which, contrastly, approaches zero surface area: http://en.wikipedia.org/wiki/Sierpinski_triangle

not constrast; it's analogous, in lower dimension.

Sponge has infinite area and 0 volume. Sierpinski's triangle has infinite perimiter and 0 area.