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Future highly renewable energy systems might require substantial storage deployment. At the current stage, the technology portfolio of dominant storage options is limited to pumped-hydro storage and Li-Ion batteries. It is uncertain which storage design will be able to compete with these options. Considering Europe as a case study, we derive the cost and efficiency requirements of a generic storage technology, which we refer to as storage-X, to be deployed in the cost-optimal system. This is performed while including existing pumped-hydro facilities and accounting for the competition from stationary Li-ion batteries, flexible generation technology, and flexible demand in a highly renewable sector-coupled energy system. Based on a sample space of 724 storage configurations, we show that energy capacity cost and discharge efficiency largely determine the optimal storage deployment, in agreement with previous studies. Here, we show that charge capacity cost is also important due to its impact on renewable curtailment. A significant deployment of storage-X in a cost-optimal system requires (a) discharge efficiency of at least 95%, (b) discharge efficiency of at least 50% together with low energy capacity cost (10EUR/kWh), or (c) discharge efficiency of at least 25% with very low energy capacity cost (2EUR/kWh). Comparing our findings with seven emerging technologies reveals that none of them fulfill these requirements. Thermal Energy Storage (TES) is, however, on the verge of qualifying due to its low energy capacity cost and concurrent low charge capacity cost. Exploring the space of storage designs reveals that system cost reduction from storage-X deployment can reach 9% at its best, but this requires high round-trip efficiency (90%) and low charge capacity cost (35EUR/kW).