Benefiting from the high resolution in beamspace, millimeter wave (mmwave) communication has been regarded as a high-accuracy localization solution, where the location information is embedded in the channel via angle and time delay, for example. In this paper, to locate a user equipment (UE) and scatterers, we present the localization model in mmwave communications as a compressed sensing assisted channel estimation problem, which is solved using a proposed two-stage channel estimation based localization scheme. During the first stage, a sparse Bayesian learning (SBL) algorithm is operated to attain a coarse estimation. Then during the second stage, a multi-stage grid refinement assisted fine estimation is achieved by a distributed compressed sensing simultaneous orthogonal matching pursuit (DCS-SOMP) algorithm. Moreover, in our approach, the few-bit analog to digital converters (ADCs) are utilized by the receiver of UE so as to attain a good trade-off among performance, complexity and energy-efficiency. Finally, the performance of channel estimation and positioning is comprehensively investigated and compared. It can be shown that our proposed two-stage approach is capable of achieving centimeter-level accuracy with the required number of quantization bits of ADCs less than four.