An accurate characterization of the known exoplanet population is key to understand the origin and evolution of planetary systems. The determination of true planetary masses through the radial velocity (RV) method is expected to experience a great improvement thanks to the availability of ultra-stable echelle spectrographs. We took advantage of the extreme precision of the new- generation echelle spectrograph ESPRESSO to characterize the transiting planetary system orbiting the G2V star K2-38 located at 194pc from the Sun with V~11.4. This system is particularly interesting because it could contain the densest planet detected to date. 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 K2-38b and K2-38c with Pb=4.01593+/-0.00050d and Pc=10.56103+/-0.00090d, respectively. Using 43 ESPRESSO high-precision radial velocity measurements taken over 8 months along with the 14 previously published HIRES RV measurements, we modeled the orbits of the two planets through a Markov Chain Monte Carlo (MCMC) analysis, significantly improving their mass measurements. Using ESPRESSO spectra we derived the stellar paremeters, Teff=5731+/-66K, logg=4.38+/-0.11dex, and [Fe/H]=0.26+/-0.05dex, and thus the mass and radius of K2-38, Ms=1.03^+0.04^-0.02_M{sun} and Rs=1.06^+0.09^-0.06_R_[sun}. We determined new values for the planetary properties of both planets. We characterized K2-38b as a super-Earth with Rp=1.54+/-0.14R{earth} and Mp=7.3^+1.1^-1.0_M_{earth}, and K2-38c as a sub-Neptune with Rp=2.29+/-0.26R{earth} and Mp=8.3+/-1.3M{earth}. Combining the radius and mass measurements, we derive a mean density of rho_p=11.0^+4.1^-2.8_g/cm^3^ for K2-38b and rho_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 an ice-rich model. 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 0.25-3M_Jup planet or stellar activity.