Nevanlinna domains are an important class of bounded simply connected domains in the complex plane; they are images of the unit disc under mappings by univalent functions belonging to model spaces (i.e. the subspaces of the Hardy space H² invariant with respect to the backward shift operator). Nevanlinna domains play a crucial role in recent progress in problems of uniform approximation of functions on compact sets in C by polyanalytic polynomials and polyanalytic rational functions. We give a complete solution to the following problem posed in the early 2000s: how large (in the sense of dimension) can be the boundaries of Nevanlinna domains? We establish the existence of Nevanlinna domains with large boundaries. In particular, these domains can have boundaries of positive planar measure. The sets of accessible points can be of any Hausdorff dimension between 1 and 2. As a quantitative counterpart of these results, we construct rational functions univalent in the unit disc with extremely long boundaries for a given amount of poles.