Context. The Transiting Exoplanet Survey Satellite (TESS) is revolutionising the search for planets orbiting bright and nearby stars. In sectors 3 and 4, TESS observed TOI-402 (TIC-120896927), a bright V = 9.1 K1 dwarf also known as HD 15337, and found two transiting signals with periods of 4.76 and 17.18 days and radii of 1.90 and 2.21 R⊕, respectively. This star was observed prior to the TESS detection as part of the radial-velocity (RV) search for planets using the HARPS spectrometer, and 85 precise RV measurements were obtained before the launch of TESS over a period of 14 yr. Aims: In this paper, we analyse the HARPS RV measurements in hand to confirm the planetary nature of these two signals. Methods: HD 15337 happens to present a stellar activity level similar to the Sun, with a magnetic cycle of similar amplitude and RV measurements that are affected by stellar activity. By modelling this stellar activity in the HARPS radial velocities using a linear dependence with the calcium activity index log(RHK'), we are able, with a periodogram approach, to confirm the periods and the planetary nature of TOI-402.01 and TOI-402.02. We then derive robust estimates from the HARPS RVs for the orbital parameters of these two planets by modelling stellar activity with a Gaussian process and using the marginalised posterior probability density functions obtained from our analysis of TESS photometry for the orbital period and time of transit. Results: By modelling TESS photometry and the stellar host characteristics, we find that TOI-402.01 and TOI-402.02 have periods of 4.75642 ± 0.00021 and 17.1784 ± 0.0016 days and radii of 1.70 ± 0.06 and 2.52 ± 0.11 R⊕ (precision 3.6 and 4.2%), respectively. By analysing the HARPS RV measurements, we find that those planets are both super-Earths with masses of 7.20 ± 0.81 and 8.79 ± 1.68 M⊕ (precision 11.3 and 19.1%), and small eccentricities compatible with zero at 2σ. Conclusions: Although having rather similar masses, the radii of these two planets are very different, putting them on different sides of the radius gap. By studying the temporal evolution under X-ray and UV (XUV) driven atmospheric escape of the TOI-402 planetary system, we confirm, under the given assumptions, that photo-evaporation is a plausible explanation for this radius difference. Those two planets, being in the same system and therefore being in the same irradiation environment are therefore extremely useful for comparative exoplanetology across the evaporation valley and thus bring constraints on the mechanisms responsible for the radius gap. Table A.1 is only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/627/A43Based on observations made with the HARPS instrument on the ESO 3.6 m telescope at La Silla Observatory under the GTO program 072.C-0488(E) and Large programs 183.C-0972(A), 192.C-0852(A), 196.C-1006 and 198.C-0836(A).