We report the confirmation of three exoplanets transiting TOI-4010 (TIC-352682207), a metal-rich K dwarf observed by the Transiting Exoplanet Survey Satellite in Sectors 24, 25, 52, and 58. We confirm these planets with the High Accuracy Radial velocity Planet Searcher for the Northern Hemisphere radial velocity observations and measure their masses with 8-12% precision. TOI-4010 b is a sub-Neptune (P = 1.3 days, ${R}_{p}={3.02}_{-0.08}^{+0.08}\,{R}_{\oplus }$ , ${M}_{p}={11.00}_{-1.27}^{+1.29}\,{M}_{\oplus }$ ) in the hot-Neptune desert, and is one of the few such planets with known companions. Meanwhile, TOI-4010 c (P = 5.4 days, ${R}_{p}={5.93}_{-0.12}^{+0.11}\,{R}_{\oplus }$ , ${M}_{p}={20.31}_{-2.11}^{+2.13}\,{M}_{\oplus }$ ) and TOI-4010 d (P = 14.7 days, ${R}_{p}={6.18}_{-0.14}^{+0.15}\,{R}_{\oplus }$ , ${M}_{p}={38.15}_{-3.22}^{+3.27}\,{M}_{\oplus }$ ) are similarly sized sub-Saturns on short-period orbits. Radial velocity observations also reveal a super-Jupiter-mass companion called TOI-4010 e in a long-period, eccentric orbit (P ~ 762 days and e ~ 0.26 based on available observations). TOI-4010 is one of the few systems with multiple short-period sub-Saturns to be discovered so far.