Context. K2-36 is a K dwarf orbited by two small (R_b = 1.43 ± 0.08 R_⊕ and R_c = 3.2 ± 0.3 R_⊕), close-in (a_b = 0.022 au and a_c = 0.054 au) transiting planets discovered by the Kepler/K2 space observatory. They are representatives of two distinct families of small planets (R_p < 4 R_⊕) recently emerged from the analysis of Kepler data, with likely a different structure, composition and evolutionary pathways. Aims. We revise the fundamental stellar parameters and the sizes of the planets, and provide the first measurement of their masses and bulk densities, which we use to infer their structure and composition. Methods. We observed K2-36 with the HARPS-N spectrograph over ~3.5 yr, collecting 81 useful radial velocity measurements. The star is active, with evidence for increasing levels of magnetic activity during the observing time span. The radial velocity scatter is ~17 m s⁻¹ due to the stellar activity contribution, which is much larger that the semi-amplitudes of the planetary signals. We tested different methods for mitigating the stellar activity contribution to the radial velocity time variations and measuring the planet masses with good precision. Results. We find that K2-36 is likely a ~1 Gyr old system, and by treating the stellar activity through a Gaussian process regression, we measured the planet masses m_b = 3.9 ± 1.1 M_⊕ and m_c = 7.8 ± 2.3 M_⊕. The derived planet bulk densities ρ_b = 7.2_−2.1^+2.5 g cm⁻³ and ρ_c = 1.3_−0.5^+0.7 g cm⁻³ point out that K2-36 b has a rocky, Earth-like composition, and K2-36 c is a low-density sub-Neptune. Conclusions. Composed of two planets with similar orbital separations but different densities, K2-36 represents an optimal laboratory for testing the role of the atmospheric escape in driving the evolution of close-in, low-mass planets after ~1 Gyr from their formation. Due to their similarities, we performed a preliminary comparative analysis between the systems K2-36 and Kepler-36, which we deem worthy of a more detailed investigation.