A simple mechanical model of planar fibrous materials with mesoscopic disorder is introduced and analyzed. In this scalar model a shear modulus controls the stress transfer in the transverse direction. The system is studied using the effective medium approximation and computer simulations; the comparison between them is quite favorable. In the disorder-controlled regime the stress-strain relation, the number of broken cells at the onset of crack propagation, and the length of the final crack scale with the system size as L2, L1.7, and L, respectively. The mechanical properties are controlled by the interplay between disorder and shear modulus, which is studied in detail.