One of the biggest and most formidable questions that we have to face in loop gravity has to do with the relation to the continuum: how does a continuous spacetime geometry arise from such discrete elements as spin-networks? In my talk, I will address this question within the simplest quantum theory of gravity, namely three-dimensional euclidean gravity with vanishing cosmological constant in regions with boundaries at finite distance. I will argue, in particular, that the discrete spectra for the geometric boundary observables that we find in loop quantum gravity can be understood from the quantization of a conformal boundary field theory in the continuum without ever introducing spin networks or triangulations of space. At a technical level, the starting point is the Hamiltonian formalism for general relativity in regions with boundaries at finite distance. At these finite boundaries, I will choose specific conformal boundary conditions that are derived from a boundary field theory for a SU(2) boundary spinor, which is minimally coupled to the spin connection in the bulk. The resulting boundary equations of motion define a boundary CFT with vanishing central charge. I will then quantize this boundary field theory and show that the length of a one-dimensional cross section of the boundary has a discrete spectrum. The lowest eigenstate of length is the continuum analogue of the Ashtekar-Lewandowski vacuum, and for every different cross section, there is a different such Fock vacuum corresponding to different superselection sectors of the boundary field theory. In addition, I will introduce a new case of coherent states and study the quasi-local observables that generate the quasi-local Virasoro algebra.