Network localization and synchronization (NLS) is a paradigm that considers joint inference of positions and clock parameters in a network consisting of completely asynchronous nodes. NLS has the potential to achieve significant performance gains in terms of localization and synchronization accuracy. In this paper, we derive fundamental performance limits of NLS by considering a problem formulation in the non-Bayesian inference framework, in which the waveforms received by different nodes in the network are considered as measurements. We perform equivalent Fisher information analysis to obtain bounds on the accuracy of NLS, and our results reveal how physical parameters and signal departure times affect the inference performance. The analytical results are verified by simulations based on a realistic channel model that takes spatial consistency into consideration.