The reliability study of k-out-of-n systems is of interest both from theoretical and practical points of view. Applications of such models can be seen in many real-world phenomena, including telecommunication, transmission, transportation, manufacturing, and services. A probabilistic study of a real-world k-out-of-n system often helps to develop an optimal strategy for maintaining high system-level reliability. There are many investigations devoted to the reliability-centric analysis of such systems. We consider a mathematical model of a repairable k-out-of-n system that works until k of its n components have failed. During the system's life cycle, its components are repaired with the help of a single repair facility. It is supposed that the components' lifetimes have an exponential distribution and their repair times have a general distribution. The proposed model is intended to be applied to the description of operation of unmanned rotorcraft high-altitude platforms and to be validated with the help of an experimental prototype. For the considered system, we propose an algorithm for calculation of the reliability function, and for special cases, k = 2 and k = 3, its closed-form representation is given. A numerical investigation is performed for special cases. The obtained results are a first step toward the sensitivity analysis of reliability characteristics of k-out-of-n systems to the shape of the repair time distributions of their components.