Higher eukaryotic chromosomes are organized into topologically constrained functional domains; however, the molecular mechanisms required to sustain these complex interphase chromatin structures are unknown. A stable matrix underpinning nuclear organization was hypothesized, but the idea was abandoned as more dynamic models of chromatin behavior became prevalent. Here, we report that scaffold attachment factor A (SAF-A), originally identified as a structural nuclear protein, interacts with chromatin-associated RNAs (caRNAs) via its RGG domain to regulate human interphase chromatin structures in a transcription-dependent manner. Mechanistically, this is dependent on SAF-A’s AAA+ ATPase domain, which mediates cycles of protein oligomerization with caRNAs, in response to ATP binding and hydrolysis. SAF-A oligomerization decompacts large-scale chromatin structure while SAF-A loss or monomerization promotes aberrant chromosome folding and accumulation of genome damage. Our results show that SAF-A and caRNAs form a dynamic, transcriptionally responsive chromatin mesh that organizes large-scale chromosome structures and protects the genome from instability.