The spatial problem of elasticity for a layer supported by two cylindrical supports embedded in it is solved. There are cylindrical spacers between the supports and the layer. The layer also has a cylindrical cavity parallel to the supports and the layer surfaces. The model is represented as an infinite layer with three inhomogeneities: two pipes (gaskets) and a cylindrical cavity. The materials of the layer and pipes are elastic and homogeneous. To obtain highly accurate results, the spatial problem of elasticity is solved using the generalized Fourier method. The layer and inhomogeneities are considered in the Cartesian and local cylindrical coordinate systems. Based on the Lamé equations and boundary conditions, an infinite system of integro-algebraic equations is formed, which is subsequently reduced to a finite system of linear algebraic equations for further numerical analysis. With the order of the system of equations (Fourier series terms) m = 4, the accuracy of the boundary conditions was 10–4 for stress values from 0 to 1 and increases with the increase in the order of the system of equations. The analysis of the stress state was carried out at different physical and mechanical characteristics of cylindrical gaskets. The results obtained show that when the material of the cylindrical gasket changes, the stress state increases significantly on the inner surfaces of the pipes and at the upper boundary of the layer. In addition, in some areas, the stress state changes sign to the opposite. The proposed method can be used to obtain high-precision results when designing parts of machines and mechanisms whose model is a layer on cylindrical embedded supports. The obtained numerical results can be used to predict the geometric parameters of real structures.