We use Monte Carlo methods to investigate the end-to-end distance distribution and entropic elasticity of self-avoiding walks in a three-dimensional half-space with both ends adsorbed on the limiting surface, The obtained distributions are well described by the Redner-des Cloizeaux (RdC) ansatz q(x)=Cx(theta) exp(-(Kx)(t)), x being the rescaled length. Using the recent solution of the junction affine model for networks of RdC springs we apply the results to the cytoskeleton of the red blood cell (RBC), a two-dimensional network of spectrin molecules which is attached to the inner surface of the erythrocyte membrane. The shear moduli predicted for a noninteracting surface are in close agreement with simulation results by Boal for a bead-spring model of the spectrin network. Moreover, we calculate stress-strain relations for finite deformations. In particular for a network which is fully adsorbed on the bilayer we find a strongly nonlinear elastic response. Our results suggest that the elastic properties of RBCs cannot be obtained within the usual Gaussian models and depend sensitively on the degree of adsorption of the spectrin network.