  Small low-mass stars are favourable targets for the detection of rocky
habitable planets. In particular, planetary systems in the solar neighbourhood
are interesting and suitable for precise characterisation. The Red Dots
campaigns seek to discover rocky planets orbiting nearby low-mass stars. The
2018 campaign targeted GJ 1061, which is the 20$^{\rm th}$ nearest star to the
Sun. For three consecutive months we obtained nightly, high-precision radial
velocity measurements with the HARPS spectrograph. We analysed these data
together with archival HARPS data. We report the detection of three planet
candidates with periods of $3.204\pm 0.001$, $6.689\pm 0.005$ and $13.03\pm
0.03$ days, which is close to 1:2:4 period commensurability. After several
considerations related to the properties of the noise and sampling, we conclude
that a 4$^{\rm th}$ signal is most likely explained by stellar rotation,
although it may be due to a planet. The proposed three-planet system (and the
potential four-planet solution) is long-term dynamically stable. Planet-planet
gravitational interactions are below our current detection threshold. The
minimum masses of the three planets range from $1.4\pm 0.2$ to $1.8\pm 0.3$
M$_\oplus$. Planet d, with $m \sin i = 1.68\pm0.25$ M$_\oplus$, receives a
similar amount of energy as Earth receives from the Sun. Consequently it lies
within the liquid-water habitable zone of the star and has a similar
equilibrium temperature to Earth. GJ 1061 has very similar properties to
Proxima Centauri but activity indices point to lower levels of stellar
activity.
