We associate to each positroid variety in the Grassmannian $\mathrm{Gr}(k,n)$ a polyhedral complex, which we call a fence complex. Fence complexes consist of unions of faces of the Gelfand-Tsetlin polytope $P_{k,n}$ associated to a fundamental weight $ω_k$. We show that these fence complexes are homeomorphic to closed balls. Furthermore, they endow the Gelfand-Tsetlin polytope with the structure of a regular CW complex, giving a polyhedral complex presentation of the regular CW complex structure on $\mathrm{Gr}(k,n)_{\geq 0}$. We also show that the Ehrhart polynomial of a fence complex equals the Hilbert polynomial of the associated positroid variety. We prove that under the Sturmfels-Gonciulea-Lakshmibai degeneration of $\mathrm{Gr}(k,n)$ to the toric variety of the Gelfand-Tsetlin polytope, positroid varieties degenerate to the reduced union of toric varieties corresponding to their fence complexes. As an application, we classify when positroid varieties contained inside hook Schubert varieties are arithmetically Gorenstein. We also derive a recursive character formula for cyclic Demazure modules, which we show is equivalent to a formula of Almousa, Gao and Huang.