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 $\tan \alpha$ (the conditions are equivalent) are in the "constructible field," the smallest extension of the rationals in which the square root of any positive element is still in the field. One might also 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
See also Marvin Jay Greenberg, "Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries," The American Mathematical Monthly (an M.A.A. journal), Volume 117, number 3, March 2010, pages 198-219. I have a pdf of that as well if anyone cannot find it.
There is a bit of a story. The results on constructibility in $H^2$ were in a string of papers in Russian and Ukrainian in the 1930's and 1940's. I found, and used, the simple conclusions. I later sent my paper to Greenberg, so that material is in the M.A.A. paper mentioned and in the fourth edition of his book. Meanwhile, Robin Hartshorne (yes, that Hartshorne) heard of this result from Marvin and came up with his own proof using the Hilbert Field of Ends formalism, expressing regret that such a pretty result did not make it into his own book on the subject, "Geometry: Euclid and Beyond."