Exoplanet surveys have shown that systems with multiple low-mass planets on compact orbits are common. Except for a few cases, however, the masses of these planets are generally unknown. At the very end of the main sequence, host stars have the lowest mass and hence offer the largest reflect motion for a given planet. In this context, we monitored the low-mass (0.13 M⊙) M dwarf YZ Cet (GJ 54.1, HIP 5643) intensively and obtained radial velocities and stellar-activity indicators derived from spectroscopy and photometry, respectively. We find strong evidence that it is orbited by at least three planets in compact orbits (POrb = 1.97, 3.06, 4.66 days), with the inner two near a 2:3 mean-motion resonance. The minimum masses are comparable to the mass of Earth (M sin i = 0.75 ± 0.13, 0.98 ± 0.14, and 1.14 ± 0.17 M⊕), and they are also the lowest masses measured by radial velocity so far. We note the possibility for a fourth planet with an even lower mass of M sin i = 0.472 ± 0.096 M⊕ at POrb = 1.04 days. An n-body dynamical model is used to place further constraints on the system parameters. At 3.6 parsecs, YZ Cet is the nearest multi-planet system detected to date. Based on observations made with the HARPS instrument on the ESO 3.6 m telescope under the program IDs 180.C-0886(A), 183.C-0437(A), and 191.C-0873(A) at Cerro La Silla (Chile).Radial velocity data (Table B.4) are only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (http://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/605/L11