Hybrid energy storage systems (HESS) are considered for use in renewable residential DC microgrids. This architecture is shown as a technically feasible solution to deal with the stochasticity of renewable energy sources, however, the complexity of its design and management increases inexorably. To address this problem, this paper proposes a fuzzy logic-based energy management system (EMS) for use in grid-connected residential DC microgrids with HESS. It is a hydrogen-based HESS, composed of batteries and multi-stack fuel cell system. The proposed EMS is based on a multivariable and multistage fuzzy logic controller, specially designed to cope with a multi-objective problem whose solution increases the microgrid performance in terms of efficiency, operating costs, and lifespan of the HESS. The proposed EMS considers the power balance in the microgrid and its prediction, the performance and degradation of its subsystems, as well as the main electricity grid costs. This article assesses the performance of the developed EMS with respect to three reference EMSs present in the literature: the widely used dual-band hysteresis and two based on multi-objective model predictive control. Simulation results show an increase in the performance of the microgrid from a technical and economic point of view.