Context. Multiplanet systems are excellent laboratories to test planet formation models as all planets are formed under the same initial conditions. In this context, systems transiting bright stars can play a key role, since planetary masses, radii, and bulk densities can be measured. Aims: GJ 9827 (K2-135) has recently been found to host a tightly packed system consisting of three transiting small planets whose orbital periods of 1.2, 3.6, and 6.2 days are near the 1:3:5 ratio. GJ 9827 hosts the nearest planetary system ( 30 pc) detected by NASA's Kepler or K2 space mission. Its brightness (V = 10.35 mag) makes the star an ideal target for detailed studies of the properties of its planets. Methods: Combining the K2 photometry with high-precision radial-velocity measurements gathered with the FIES, HARPS, and HARPS-N spectrographs we revised the system parameters and derive the masses of the three planets. Results: We find that GJ 9827 b has a mass of Mb = 3.69-0.46+0.48 M⊕ and a radius of Rb = 1.58-0.13+0.14 R⊕, yielding a mean density of ρb = 5.11-1.27+1.74 g cm-3. GJ 9827 c has a mass of Mc = 1.45-0.57+0.58 M⊕, radius of Rc = 1.24-0.11+0.11 R⊕, and a mean density of ρc = 4.13-1.77+2.31 g cm-3. For GJ 9827 d, we derive Md = 1.45-0.57+0.58 M⊕, Rd = 1.24-0.11+0.11 R⊕, and ρd = 1.51-0.53+0.71 g cm-3. Conclusions: GJ 9827 is one of the few known transiting planetary systems for which the masses of all planets have been determined with a precision better than 30%. This system is particularly interesting because all three planets are close to the limit between super-Earths and sub-Neptunes. The planetary bulk compositions are compatible with a scenario where all three planets formed with similar core and atmosphere compositions, and we speculate that while GJ 9827 b and GJ 9827 c lost their atmospheric envelopes, GJ 9827 d maintained its primordial atmosphere, owing to the much lower stellarirradiation. This makes GJ 9827 one of the very few systems where the dynamical evolution and the atmosphericescape can be studied in detail for all planets, helping us to understand how compact systems form and evolve. Based on observations made with (a) the ESO-3.6 m telescope at La Silla Observatory under program ID 099.C-0491 and 0100.C-0808; (b) the Italian Telescopio Nazionale Galileo operated on the island of La Palma by the Fundación Galileo Galilei of the Istituto Nazionale di Astrofisica; (c) the Nordic Optical Telescope, operated by the Nordic Optical Telescope Scientific Association at the Observatorio del Roque de los Muchachos.