Multiplanet systems are valuable arenas for investigating exoplanet architectures and comparing planetary siblings. TOI-1246 is one such system, with a moderately bright K dwarf (V = 11.6, K = 9.9) and four transiting sub-Neptunes identified by TESS with orbital periods of 4.31, 5.90, 18.66, and 37.92 days. We collected 130 radial velocity observations with Keck/HIRES and TNG/HARPS-N to measure planet masses. We refit the 14 sectors of TESS photometry to refine planet radii (2.97 ± 0.06 R ⊕, 2.47 ± 0.08 R ⊕, 3.46 ± 0.09 R ⊕, and 3.72 ± 0.16 R ⊕) and confirm the four planets. We find that TOI-1246 e is substantially more massive than the three inner planets (8.1 ± 1.1 M ⊕, 8.8 ± 1.2 M ⊕, 5.3 ± 1.7 M ⊕, and 14.8 ± 2.3 M ⊕). The two outer planets, TOI-1246 d and TOI-1246 e, lie near to the 2:1 resonance (P e/P d = 2.03) and exhibit transit-timing variations. TOI-1246 is one of the brightest four-planet systems, making it amenable for continued observations. It is one of only five systems with measured masses and radii for all four transiting planets. The planet densities range from 0.70 ± 0.24 to 3.21 ± 0.44 g cm-3, implying a range of bulk and atmospheric compositions. We also report a fifth planet candidate found in the RV data with a minimum mass of 25.6 ± 3.6 M ⊕. This planet candidate is exterior to TOI-1246 e, with a candidate period of 93.8 days, and we discuss the implications if it is confirmed to be planetary in nature.