We create tailored microstructures, consisting of complexes of lipid membranes with self-assembled biopolymer shells, to study the fundamental properties and interactions of these basic components of living cells. We measure the mechanical response of these artificial structures at the micrometer scale, using optical tweezers and single-particle tracking. These systems exhibit rich dynamics that illustrate the viscoelastic character of the quasi-two-dimensional biopolymer network. We present a theoretical model relating the rheological properties of these membranes to the observed dynamics.