The equilibrium current fluctuations (thermal noise) for a 2D degenerate electron gas bounded in a narrow channel have been calculated within the semiclassical Boltzmann-equation approach. The analytical formula for the autocorrelation function has been derived for the completely degenerate case. The associated noise spectrum has a non-Lorentzian shape with decreased zero-frequency plateau and smeared geometrical resonances at high frequencies caused by the restriction on electron motion in the transverse direction (classical size effect). Measurements of the size-dependent noise spectrum would give additional information about the edge scattering of electrons.