While actin bundles are used by living cells for structural fortification, the microscopic origin of the elasticity of bundled networks is not understood. Here, we show that above a critical concentration of the actin binding protein fascin, a solution of actin filaments organizes into a pure network of bundles. While the elasticity of weakly cross-linked networks is dominated by the affine deformation of tubes, the network of bundles can be fully understood in terms of nonaffine bending undulations.