The aim of this work was to provide a guidance to the prediction and design of high-entropy alloys with good performance. New promising compositions of refractory high-entropy alloys with the desired phase composition and mechanical properties (yield strength) have been predicted using a combination of machine learning, phenomenological rules and CALPHAD modeling. The yield strength prediction in a wide range of temperatures (20–800 °C) was made using a surrogate model based on a support-vector machine algorithm. The yield strength at 20 °C and 600 °C was predicted quite precisely (the average prediction error was 11% and 13.5%, respectively) with a decrease in the precision to slightly higher than 20% at 800 °C. An Al13Cr12Nb20Ti20V35 alloy with an excellent combination of ductility and yield strength at 20 °C (16.6% and 1295 MPa, respectively) and at 800 °C (more 50% and 898 MPa, respectively) was produced based on the prediction.