We consider a general multi-period repositioning problem in vehicle-sharing networks such as bicycle-sharing systems, free-float car-sharing systems, and autonomous mobility-on-demand systems. This problem is subject to uncertainties along multiple dimensions - including demand, travel time, and repositioning duration - and faces several operational constraints such as the service level and cost budget. We propose a robustness optimization model to tackle these uncertainties; thus we aim to satisfy operational constraints under a reference distribution yet also to protect against ambiguity in the true distribution. This paper is the first, as far as we know, to incorporate various time-dependent uncertainties. We then reformulate the model and efficiently obtain solutions by solving a sequence of mixed-integer linear optimization problems. Extensive simulation studies demonstrate that our model yields remarkable performance in various settings and is computationally scalable. We find that our model, when compared to such benchmarks as "fluid-based optimization", achieves the highest average service level for a given repositioning cost budget; it also is robust to adverse circumstances (i.e., its worst-case service level is also the highest).