Mobile Edge Computing (MEC) (a.k.a. fog computing) has recently emerged to enable low-latency and location-aware data processing at the edge of mobile networks. Providing grid power supply in support of MEC, however, is costly and even infeasible, thus mandating on-site renewable energy as a major or even sole power supply in many scenarios. Nonetheless, the high intermittency and unpredictability of energy harvesting creates many new challenges of performing effective MEC. In this paper, we develop an algorithm called GLOBE that performs joint geographical load balancing (GLB) (for computation workload) and admission control (for communication data traffic), for optimizing the system performance of a network of MEC-enabled base stations. By leveraging the Lyapunov optimization with perturbation technique, GLOBE operates online without requiring future system information and addresses significant challenges caused by battery state dynamics and energy causality constraints. We prove that GLOBE achieves a close-to-optimal system performance compared to the offline algorithm that knows full future information, and present a critical tradeoff between battery capacity and system performance. Simulation results validate our analysis and demonstrate the superior performance of GLOBE compared to benchmark algorithms.