This problem was posed in Berkeley about 1953 by Besikovitch (Абрам Самойлович Безикович) when he was visiting. I think that the following solution was also his.
If the curve is not rectifiable the result is vacuously true. If it is rectifiable choose two points, X and Y, on the curve so that the two resulting portions of the curve are the same length. If X and Y are antipodal we are done. Consider the unique great circle thru X and Y and the point Z on that circle midway between X and Y. Consider the great circle C that is 90 degrees from the point Z. (Z is ambiguous but C is not.)
By hypothesis the curve meets C. Choose a point W where the curve meets C. Half of the original curve is now divided into portions we will call XtoW and WtoY. Consider the point X' which is the reflection of X in the plane of C. X' is antipodal to Y. Consider a curve that is the reflection of XtoW in C’s plane and call it X'toW. X'toW is congruent to XtoW and thus the same length. Adjoining X'toW to WtoY gives a curve that connects antipodal points and is thus half as long, at least, as a great circle. But that curve is just as long as the curve from X to Y thru W. That curve was just half the length of the original however. QED.