In this article, we present our vision for how extremely large aperture arrays (ELAAs), equipped with hundreds or thousands of antennas, can play a major role in future 6G networks by enabling a remarkable increase in data rates through massive spatial multiplexing to both a single user and many simultaneous users. Specifically, with the quantum leap in the array aperture size, the users will be in the so-called radiative near-field region of the array, where previously negligible physical phenomena dominate the propagation conditions and give the channel matrices more favorable properties. This article presents the foundational properties of communication in the radiative near-field region and then exemplifies how these properties enable two unprecedented spatial multiplexing schemes: depth-domain multiplexing of multiple users and angular multiplexing of data streams to a single user. We also highlight research challenges and open problems that require further investigation.