Three-dimensional confinement allows semiconductor quantum dots to exhibit size-tunable electronic and optical properties that enable a wide range of opto-electronic applications from displays, solar cells and bio-medical imaging to single-electron devices. Additional modalities such as spin and valley properties in monolayer transition metal dichalcogenides provide further degrees of freedom requisite for information processing and spintronics. In nanostructures, however, spatial confinement can cause hybridization that inhibits the robustness of these emergent properties. Here, we show that laterally-confined excitons in monolayer MoS2 nanodots can be created through top-down nanopatterning with controlled size tunability. Unlike chemically-exfoliated monolayer nanoparticles, the lithographically patterned monolayer semiconductor nanodots down to a radius of 15 nm exhibit the same valley polarization as in a continuous monolayer sheet. The inherited bulk spin and valley properties, the size dependence of excitonic energies, and the ability to fabricate MoS2 nanostructures using semiconductor-compatible processing suggest that monolayer semiconductor nanodots have potential to be multimodal building blocks of integrated optoelectronics and spintronics systems.