We discuss general relativity in 2 + 1 dimensions with vanishing cosmological constant and in absence of matter. The phase space is identified with the cotangent bundle of Teichmüller space. We show that certain gauge invariant length observables in space-time arise as derivatives of geometric functions on Teichmüller space. The Poisson structure of these observables is established. After this we discuss the geometric quantisation of the reduced phase space. The length observables are recognised as derivative operators on wave functions on Teichmüller space. The spectrum of two of these observables is calculated. The first observable, which measures a space-like distance, turns out to have the whole real line as its spectrum. The second observable, which measures a time-like distance, is quantized, but the eigenvalue separation varies between zero and the Planck length depending on the sector of phase space. Finally we relate our results to claims made in the literature.