The mechanical properties of single-phase (Hf,Zr,Ti,Ta,Nb)C high-entropy carbide (HEC) ceramics were investigated. Ceramics with relative density > 99% and an average grain size of 0.9 ± 0.3 µm were produced by a two-step process that involved carbothermal reduction at 1600°C and hot pressing at 1900°C. At room temperature, Vickers hardness was 25.0 ± 1.0 GPa at a load of 4.9 N, Young's modulus was 450 GPa, chevron notch fracture toughness was 3.5 ± 0.3 MPa·m1/2, and four-point flexural strength was 421 ± 27 MPa. With increasing temperature, flexural strength stayed above ~400 MPa up to 1800°C, then decreased nearly linearly to 318 ± 21 MPa at 2000°C and to 93 ± 10 MPa at 2300°C. No significant changes in relative density or average grain size were noted after testing at elevated temperatures. The degradation of flexural strength above 1800°C was attributed to a decrease in dislocation density that was accompanied by an increase in dislocation motion. These are the first reported flexural strengths of HEC ceramics at elevated temperatures.