Although the Earth is not a perfect sphere, a mathematical object representing its surface (a geoid) is homeomorphic to S2. Therefore, the theorem applies.

Except for the fact that, after we've located1 the aforementioned two points, and after transforming back to the geoid, the points probably won't be exactly antipodal anymore.

ariels says "One way to do it is to show that the Earth is star convex around its centre. Define a map R^3\{0} -> R^3 by x -> x/|x|. Then the map transforms the Earth says to a sphere, while preserving antipodality."

1located, somehow. We were given an existence proof, not a construction.