The superfluid density and superconducting gaps of superconducting RbFe₂As₂ have been determined as a function of temperature, magnetic field, and hydrostatic pressure by susceptibility and muon-spin spectroscopy measurements. From the data, fundamental microscopic parameters of the superconducting state like the London penetration depth λ, the gap values Δ, the upper critical field B_c2, and the Ginzburg-Landau parameter κ have been obtained. In accordance with earlier measurements the ratio of the superfluid density n_s α λ^–2 to the superconducting transition temperature T_c = 2.52(2) K at ambient pressure is found to be much higher in the strongly hole-overdoped RbFe₂As₂ than in high-T_c Fe-based and other unconventional superconductors. As a function of pressure, T_c strongly decreases, at the rate of T_c/dp = –1.32 K GPa^–1, i.e., it is reduced by 52% at p = 1 GPa. The temperature dependence of n_s is best described by a two-gap s-wave model, with both superconducting gaps being decreased by hydrostatic pressure until the smaller gap completely disappears at p = 1 GPa.