Well, the earth's surface is certainly something like **S**^{2}. Take any two real valued functions which vary
continously as a function of their position on the surface of the earth (say
temperature and atmospheric pressure). The theorem says that there exists
antipodal points on the earth's surface where these two quantities have the
same values (thus the temperature and atmospheric pressure at both places on
the earth agree). Kinda slick, I think.
I mentioned this particular example with temperature and pressure to someone
I know who does atmospheric modelling and dynamics. They thought it so
ludicrous that their reaction was, "You mathematicians have no idea what
you're talking about!" I took this to mean that in their modelling they used
discontinuous functions of temperature and pressure ;-)

*Andrew D. Lewis (andrew at mast.queensu.ca)*