Augmenting ground-level communications with flying networks, such as the high-altitude platform system (HAPS), is among the major innovative initiatives of the next generation of wireless systems (6G). Given HAPS quasi-static positioning at the stratosphere, HAPS-to-ground and HAPS-to-air connectivity frameworks are expected to be prolific in terms of data acquisition and computing, especially given the mild weather and quasi-constant wind speed characteristics of the stratospheric layer. This paper explores the opportunities stemming from the realization of cloud-enabled HAPS in the context of telecommunications applications and services. The paper first advocates for the potential physical advantages of deploying HAPS as flying data-centers, also known as super-macro base stations. The paper then describes various cloud services that can be offered from the HAPS and the merits that can be achieved by this integration, such as enhancing the quality, speed, and range of the offered services. The proposed services span a wide range of fields, including satellites, Internet of Things (IoT), ad hoc networks (such as sensor; vehicular; and aerial networks), gaming, and social networks. For each service, the paper illustrates the methods that would be used by cloud providers to offload the service data to the HAPS and enable the cloud customers to consume the service. The paper further sheds light on the challenges that need to be addressed for realizing practical cloud-enabled HAPS, mainly, those related to high energy, processing power, quality of service (QoS), and security considerations. Finally, the paper discusses some open issues on the topic, namely, HAPS mobility and message routing, HAPS security via blockchain and machine learning, artificial intelligence-based resource allocation in cloud-enabled HAPS, and integration with vertical heterogeneous networks.