An elastic space with an absolutely rigid inclusion whose shape is different from the customarily examined ellipsoidal one is considered. It is assumed that the inclusion is in the form of a thin torus whose cross section is an arbitrary two-dimensional domain. Using a method for investigating the solutions of elliptic boundary value problems in singularly perturbed domains, the asymptotic form of the stress-strain state, and also the asymptotic behavior of important characteristics such as the elastic and polarization capacitance tensors are determined.