This article presents the Templet Web cloud service. The service is designed for high-performance scientific computing automation. The use of high-performance technology is specifically required by new fields of computational science such as data mining, artificial intelligence, machine learning, and others. Cloud technologies provide a significant cost reduction for high-performance scientific applications. The main objectives to achieve this cost reduction in the Templet Web service design are: (a) the implementation of “on-demand” access; (b) source code deployment management; (c) high-performance computing programs development automation. The distinctive feature of the service is the approach mainly used in the field of volunteer computing, when a person who has access to a computer system delegates his access rights to the requesting user. We developed an access procedure, algorithms, and software for utilization of free computational resources of the academic cluster system in line with the methods of volunteer computing. The Templet Web service has been in operation for five years. It has been successfully used for conducting laboratory workshops and solving research problems, some of which are considered in this article. The article also provides an overview of research directions related to service development.