Nuclear chromosomes transcribe far more RNA than required to encode protein. Here we investigate whether non-coding RNA broadly contributes to cytological-scale chromosome territory architecture. We develop a procedure that depletes soluble proteins, chromatin, and most nuclear RNA from the nucleus but does not delocalize XIST, a known architectural RNA, from an insoluble chromosome "scaffold." RNA-seq analysis reveals that most RNA in the nuclear scaffold is repeat-rich, non-coding, and derived predominantly from introns of nascent transcripts. Insoluble, repeat-rich (C0T-1) RNA co-distributes with known scaffold proteins including scaffold attachment factor A (SAF-A), and distribution of these components inversely correlates with chromatin compaction in normal and experimentally manipulated nuclei. We further show that RNA is required for SAF-A to interact with chromatin and for enrichment of structurally embedded "scaffold attachment regions" prevalent in euchromatin. Collectively, the results indicate that long nascent transcripts contribute a dynamic structural role that promotes the open architecture of active chromosome territories.