Experimentally measuring the elastic properties of a thin-walled curved flexible biological surface is difficult. One technique which may be used is the indentation of a thin sheet of material by a rigid indenter, whilst measuring the force or displacement applied to the indenter. This gives immediate information on the fracture strength of the material (from the force required to puncture), but it is also possible to determine the elastic properties by comparing the resulting force-displacement curves with a mathematical model. Mathematical studies of this process commonly assume that the elastic surface is initially flat, which is often not the case for biological membranes. As such, we previously outlined a theory for the indentation of curved isotropic, incompressible, hyperelastic membranes (with no bending stiffness), which breaks down for highly curved surfaces, as the entire membrane becomes wrinkled. Here we treat the surface as an elastic shell by including bending stiffness, ensuring that energy is required to change the shape of the shell even without stretching. The theory presented here enables curved surfaces to be considered in particular, allowing the estimation of shape- and size-independent elastic properties from indentation experiments, and is particularly relevant for biological membranes.