One of the key tasks in the study of two-dimensional nanomaterials (fullerenes, nanotubes, graphene) is to explore their mechanical properties: bending, vibrations, and stability. For the study of such problems, it is essential to construct both microscopic and macroscopic models of the deformation behavior of such materials. Based on the three-dimensional moment theory of elasticity, the moment-membrane theory of elastic cylindrical shells is constructed as a continual model of deformations of a single-layer carbon nanotube. The axisymmetric deformation of an elastic cylindrical shell is studied, and numerical results are presented. Further, a moment-membrane technical theory of elastic cylindrical shells is constructed, on the basis of which the stability of the initially axial compressed state of a carbon nanotube is studied, and the critical load value is numerically determined.