Atomically thin two-dimensional semiconductors such as MoS2 hold great promise for electrical, optical and mechanical devices and display novel physical phenomena. However, the electron mobility of mono-and few-layer MoS2 has so far been substantially below theoretically predicted limits, which has hampered efforts to observe its intrinsic quantum transport behaviours. Potential sources of disorder and scattering include defects such as sulphur vacancies in the MoS2 itself as well as extrinsic sources such as charged impurities and remote optical phonons from oxide dielectrics. To reduce extrinsic scattering, we have developed here a van der Waals heterostructure device platform where MoS2 layers are fully encapsulated within hexagonal boron nitride and electrically contacted in a multi-terminal geometry using gate-tunable graphene electrodes. Magneto-transport measurements show dramatic improvements in performance, including a record-high Hall mobility reaching 34,000 cm2 V-1 s-1 for six-layer MoS2 at low temperature, confirming that low-temperature performance in previous studies was limited by extrinsic interfacial impurities rather than bulk defects in the MoS2. We also observed Shubnikov-de Haas oscillations in high-mobility monolayer and few-layer MoS2. Modelling of potential scattering sources and quantum lifetime analysis indicate that a combination of short-range and long-range interfacial scattering limits the low-temperature mobility of MoS2.