Not an answer to the question as stated but relevant:
I don't seem to have a reference, but the quickest description of possible constructions on, say, the unit sphere, is by the angles made by intersecting curves. The constructible angles are the same as the constructible angles in the Euclidean plane. Thus the constructible lengths are those arclengths $\alpha$ for which $\cos \alpha$ or $ \sin \alpha$ or $\tan \alpha$ are in the "constructible field," the smallest extension of the rationals in which the squareroot of any positve element is still in the field. One might wish to require $\alpha \leq \pi.$ Actually, let me make that a request. If anybody knows of a reference on the constructible lengths and angles on the surface of the sphere, please let me know.
This is strictly analogous to (and presumably far, far older than) the situation in the hyperbolic plane, I will try to make a working link:
http://zakuski.math.utsa.edu/~jagy/papers/Intelligencer_1995.pdf