Technology forecasts anticipate a new era in which massive numbers of humans, machines, and things are connected to wireless networks to sense, process, act, and communicate with the surrounding environment in a real-time manner. To make the visions come true, the sixth generation (6G) wireless networks should be hyper-connected, implying that there are no constraints on the data rate, coverage, and computing. In this article, we first identify the main challenges for 6G hyper-connectivity, including terabits-per-second (Tbps) data rates for immersive user experiences, zero coverage-hole networks, and pervasive computing for connected intelligence. To overcome these challenges, we highlight key enabling technologies for 6G such as distributed and intelligence-aware cell-free massive multi-input multioutput (MIMO) networks, boundless and fully integrated terrestrial and non-terrestrial networks, and communication-aware distributed computing for computation-intensive applications. We further illustrate and discuss the hyper-connected 6G network architecture along with open issues and future research directions.