Gaussian beams for the stationary 2D Dirac equation with inhomogeneous electric and magnetic fields are constructed. Gaussian beams (GB) are such asymptotic solutions of this equation that are exponentially localized near semiclassical trajectories. To derive formulas for the GB, we found the leading term of semiclassical asymptotic solutions of this equation by elementary methods. The results are given in such a form that can be applied to another vector problems.