Both localization and synchronization of mobile nodes are of fundamental importance for wireless networks. With the emergence of full-duplex (FD) communication technology, inter-node distances and clock offsets among a set of nodes can be simultaneously obtained through only two frames of communications, thus significantly improving the efficiency of node localization and synchronization. In this paper, we propose a localization and synchronization scheme using FD radios, and characterize its performance. Our study derives the Cramér-Rao lower bounds (CRLBs) for inter-node distances and clock offsets, the former of which can be translated into the estimation error bounds for localization. Comparison to conventional frequency division duplex (FDD) or time division duplex (TDD) demonstrates the high efficiency of localization and synchronization using FD radios. Our results reveal the potential of full-duplex technology beyond data communications in future wireless networks.