Expression of the transcription factor brachyury (TBXT) is normally restricted to the embryo and its silencing is epigenetically regulated. TBXT promotes mesenchymal transition in a subset of common carcinomas, and in chordoma, a rare cancer showing notochordal differentiation, TBXT acts as a putative oncogene. We hypothesized that TBXT expression is controlled through epigenetic inhibition to promote chordoma cell death. Screening of five human chordoma cell lines revealed that pharmacological inhibition of the histone 3 lysine 27 demethylases KDM6A (UTX) and KDM6B (JMJD3) leads to cell death. This effect was phenocopied by dual genetic inactivation of KDM6A/B using CRISPR/Cas9. Inhibition of KDM6A/B with a novel compound KDOBA67 led to a genome-wide increase in repressive H3K27me3 marks with concomitant reduction in active H3K27ac, H3K9ac, and H3K4me3 marks. TBXT was a KDM6A/B target gene, and chromatin changes at TBXT following KDOBA67 treatment were associated with a reduction in TBXT protein levels in all models tested, including primary patient-derived cultures. In all models tested, KDOBA67 treatment downregulated expression of a network of transcription factors critical for chordoma survival and upregulated pathways dominated by ATF4-driven stress and pro-apoptotic responses. Blocking the AFT4 stress response did not prevent suppression of TBXT and induction of cell death, but ectopic overexpression of TBXT increased viability, therefore implicating TBXT as a potential therapeutic target of H3K27 demethylase inhibitors in chordoma. Our work highlights how knowledge of normal processes in fetal development can provide insight into tumorigenesis and identify novel therapeutic approaches.