Apparently this is one of those opinion-based questions that are destined to be closed soon. Still, I'll share my perspective.

IMHO it doesn't matter at all (for doing mathematics, at least) if you can outrun a cheetah or to come up as a winner in a wrestling match with an elephant. What is really required is an ability to concentrate and to think for an extended period of time without developing a headache, falling asleep, or going in circles. I doubt it burns any noticeable amount of calories because the best position I know for doing it is lying on the sofa. If that is the "stamina" we are talking about, I agree, but in all other respects, any physical shape that doesn't give you an obvious trouble and provides enough oxygenation to the brain would, probably, do.

Another possible meaningful interpretation is that perseverance is at least as important as brilliance. I'm not quite sure I believe in Edison's statement that "genius is one percent inspiration and 99 percent perspiration", but I certainly agree that trying to ride the flashes of insight alone often won't bring you very far.

As to the rest, it is in the human nature to show off a bit now and then, and, for a mathematician it is a little difficult to impress a non-mathematician (or even a mathematician in a different area) with his or her professional achievements, so I suspect many of us are in the habit of developing some marketable "side skills" and going into sports is certainly an option (though by no means the only one: somebody can choose to go into, say, arts instead and, if he ever becomes as famous as Weil, leave us pondering on MO if mathematics is all about the powers of imagination and the ability to reflect and transform all that we see and feel into sophisticated creations in the parallel world).

Just my two cents.