What does it mean to say that a pair of points are antipodal in a topological sphere? A pair of points are antipodal if they are diametrically opposite to each other. This definition makes perfect sense when one thinks of the unit 2-sphere centered at the origin and embedded in $R^3$; that is, the set of all points for which $x^2+y^2+z^2=1$. However, what does it mean to say that a pair of points are antipodal in a topological sphere? If this question doesn't make sense, I fail to recognize when two points are antipodal when considering, say, an ellipsoid. For example, how does one make sense of the Borsuk–Ulam theorem for the ellipsoid?

The meaning of "antipodal" depends on a choice of homeomorphism between an ellipsoid (or any other subset of $\mathbb{R}^n$ homeomorphic to a sphere) and the usual unit sphere. The Borsuk-Ulam theorem is true regardless of this choice.