Yes. the fundamental theorem is that the constructible angles in the non-Euclidean plane are exactly the constructible angle in the Euclidean plane. Lengths come from the two laws of cosines and the law of sines. In particular, length 1 is not constructible. As i recall, positive length $x$ is constructible if and only if $\sinh x$ is constructible in the Euclidean plane. 

See my 1995 Intelligencer article at http://zakuski.utsa.edu/~jagy/bib.html 


Let's see, in the fourth edition of his book, Marvin Jay Greenberg included Robin Hartshorne's proof of this, very different sort of language. 

EDIT: it seems part of the question was constructible lengths with irrational ratio. $\log 3$ and $\log 2$ should work.