A cylindrically anisotropic magnetoelectroelastic material is a special inhomogeneous anisotropic magnetoelectroelastic material. The constitutive law for any material point is the same when it is referred to a cylindrical coordinate system. An example of cylindrically anisotropic magnetoelectroelastic material is composite made of cylindrically anisotropic piezoelectric/piezomagnetic materials. In this paper an exact solution is derived for the two-dimensional problem of a circular tube or bar of cylindrically anisotropic magnetoelectroelastic material under pressuring loading by applying the Stroh formalism for a cylindrical coordinate system. The explicit expressions for the extended displacement vector and the extended traction vectors are presented. As encountered in the cases of elastic material and piezoelectric material, the stresses, electric fields, and magnetic fields at the axis of a circular rod may be infinite when the rod is subjected to a radial pressure. The existence of the singular stresses is also verified by our calculations for some magnetoelectroelastic materials.