The SPT 0311-58 system at z = 6.900 is an extremely massive structure within the reionization epoch and offers a chance to understand the formation of galaxies at an extreme peak in the primordial density field. We present 70 mas Atacama Large Millimeter/submillimeter Array observations of the dust continuum and [C II] 158 mu m emission in the central pair of galaxies and reach physical resolutions of similar to 100-350 pc, among the most detailed views of any reionization-era system to date. The observations resolve the source into at least a dozen kiloparsecsize clumps. The global kinematics and high turbulent velocity dispersion within the galaxies present a striking contrast to recent claims of dynamically cold thin-disk kinematics in some dusty galaxies just 800 Myr later at z similar to 4. We speculate that both gravitational interactions and fragmentation from massive parent disks have likely played a role in the overall dynamics and formation of clumps in the system. Each clump individually is comparable in mass to other 6 < z < 8 galaxies identified in rest-UV/optical deep field surveys, but with star formation rates elevated by a factor of similar to 3-5. Internally, the clumps themselves bear close resemblance to greatly scaled-up versions of virialized cloud-scale structures identified in low-redshift galaxies. Our observations are qualitatively similar to the chaotic and clumpy assembly within massive halos seen in simulations of high-redshift galaxies.