Rotobreathers in the chain of coupled linearly elastic rotators are analyzed. Each rotator is a particle connected by a massless elastic rod with a frictionless pivot; it has two degrees of freedom, length and angle of rotation. The rods of the rotators and the elastic bonds between the nearest rotators are linearly elastic, and the nonlinearity of the system is of a purely geometric nature. It is shown that long-lived rotobreathers can exist if the stiffness of the rods is high enough to create a relatively wide gap in the phonon spectrum of the chain. The frequency of angular rotation of the rotobreather cannot be above the optical band of the phonon spectrum and is in the spectrum gap. Generally speaking, the rotation of the rotobreather is accompanied by radial oscillations; however, one can choose such initial conditions so that the radial oscillations are minimal. Some parameters of rotobreathers with minimal radial vibrations are presented on the basis of numerical simulations. The results obtained qualitatively describe the behavior of physical systems with coupled rotators.