Aims. We aim to constrain the dynamical masses of the four confirmed transiting planets orbiting the infant (20 ± 10 Myr) star V1298 Tau (orbital periods Pb~ 24.1 d, Pc~ 8.2 d, Pd~ 12.4 d, and Pe~ 48.7 d), through a new analysis of a homogeneous radial-velocity dataset. This system is a key benchmark for testing formation and early-stage evolutionary models of planets. Methods. We analysed more than 380 high-resolution spectra collected with the HARPS-N spectrograph over a time span of 4.5 yr. To account for dominant stellar activity and transit-timing-derived planet--planet interactions, we analysed radial velocities extracted using two different approaches and combined N-body simulations with multi-dimensional Gaussian-process regression. We tested a set of spectroscopic and photometric activity diagnostics to mitigate the variability due to the stellar magnetic activity in the radial-velocity time series. Results. The mass determinations are sensitive to the choice of the radial-velocity extraction method and activity diagnostics. For some specific models, we measured masses with a significance of 3.5--4σ for planets b (0.52-0.14+0.12, 0.65-0.18+0.21, 0.67-0.19+0.16, and 0.78-0.20+0.19 MJup) and e (0.40-0.13+0.14, 0.66-0.18+0.17, and 0.71-0.22+0.18 MJup). These values are compatible with masses of giant planets and are notably higher than those recently derived from transit timing variations alone, which indicate that these are low-density `super-puffy' planets. Injection-and-recovery simulations moderately support this conclusion for planet b. For planets c and d, we primarily provide updated upper mass limits. Conclusions. Our analysis represents an exploratory study to address the complex issue of mass measurement in a multi-planetary system characterised by high levels of stellar activity. While firm conclusions cannot be drawn for V1298 Tau, in a few cases our results leave open the possibility that planet b in particular is a massive giant rather than a sub-Neptune progenitor. This dichotomy may be solved by combining radial velocities and transit time variations in a more complex photo-dynamical modelling, with a larger sample of transits than currently available.