In this work, a Fault Ride-Through control scheme for a non-isolated power topology for Hybrid Energy Storage Systems in a DC microgrid is presented. The Hybrid System is created from a Lithium-Ion Battery and a Supercapacitor Module coordinated to achieve a high-energy and high-power storage system; it is connected to a DC link to interface to the outer system. The power topology under consideration is based on the buck-boost bidirectional converter, and it is controlled through a bespoke modulation scheme in order to obtain low losses in nominal operation. The operation of the proposed control during a DC link short-circuit failure is shown as well as a modification to the standard control in order to achieve Fault Ride-Through once the fault is over. The operation of the converter is theoretically developed and it is verified through simulation and experimental validation.