We characterized the transiting planetary system orbiting the G2V star K2-38 using the new-generation echelle spectrograph ESPRESSO. We carried out a photometric analysis of the available K2 photometric light curve of this star to measure the radius of its two known planets. Using 43 ESPRESSO high-precision radial velocity measurements taken over the course of 8 months along with the 14 previously published HIRES RV measurements, we modeled the orbits of the two planets through a MCMC analysis, significantly improving their mass measurements. Using ESPRESSO spectra, we derived the stellar parameters, \(T_{\rm eff}\)=5731\(\pm\)66, \(\log g\)=4.38\(\pm\)0.11~dex, and \(Fe/H\)=0.26\(\pm\)0.05~dex, and thus the mass and radius of K2-38, \(M_{\star}\)=1.03 \(^{+0.04}_{-0.02}\)~M\(_{\oplus}\) and \(R_{\star}\)=1.06 \(^{+0.09}_{-0.06}\)~R\(_{\oplus}\). We determine new values for the planetary properties of both planets. We characterize K2-38b as a super-Earth with \(R_{\rm P}\)=1.54\(\pm\)0.14~R\(_{\rm \oplus}\) and \(M_{\rm p}\)=7.3\(^{+1.1}_{-1.0}\)~M\(_{\oplus}\), and K2-38c as a sub-Neptune with \(R_{\rm P}\)=2.29\(\pm\)0.26~R\(_{\rm \oplus}\) and \(M_{\rm p}\)=8.3\(^{+1.3}_{-1.3}\)~M\(_{\oplus}\). We derived a mean density of \(\rho_{\rm p}\)=11.0\(^{+4.1}_{-2.8}\)~g cm\(^{-3}\) for K2-38b and \(\rho_{\rm p}\)=3.8\(^{+1.8}_{-1.1}\)~g~cm\(^{-3}\) for K2-38c, confirming K2-38b as one of the densest planets known to date. The best description for the composition of K2-38b comes from an iron-rich Mercury-like model, while K2-38c is better described by a rocky model with a H2 envelope. The maximum collision stripping boundary shows how giant impacts could be the cause for the high density of K2-38b. The irradiation received by each planet places them on opposite sides of the radius valley. We find evidence of a long-period signal in the radial velocity time-series whose origin could be linked to a 0.25-3~M\(_{\rm J}\) planet or stellar activity.