The use of lithium metal anodes in solid-state batteries has emerged as one of the most promising technologies for replacing conventional lithium-ion batteries(1,2). Solid-state electrolytes are a key enabling technology for the safe operation of lithium metal batteries as they suppress the uncontrolled growth of lithium dendrites. However, the mechanical properties and electrochemical performance of current solid-state electrolytes do not meet the requirements for practical applications of lithium metal batteries. Here we report a class of elastomeric solid-state electrolytes with a three-dimensional interconnected plastic crystal phase. The elastomeric electrolytes show a combination of mechanical robustness, high ionic conductivity, low interfacial resistance and high lithium-ion transference number. The in situ-formed elastomer electrolyte on copper foils accommodates volume changes for prolonged lithium plating and stripping processes with a Coulombic efficiency of 100.0 per cent. Moreover, the elastomer electrolytes enable stable operation of the full cells under constrained conditions of a limited lithium source, a thin electrolyte and a high-loading LiNi0.83Mn0.06Co0.11O2 cathode at a high voltage of 4.5 volts at ambient temperature, delivering a high specific energy exceeding 410 watt-hours per kilogram of electrode plus electrolyte. The elastomeric electrolyte system presents a powerful strategy for enabling stable operation of high-energy, solid-state lithium batteries.