The DNA-damage response (DDR) arrests cell-cycle progression until damage is removed. DNA-damage-induced cellular senescence is associated with persistent DDR. The molecular bases that distinguish transient from persistent DDR are unknown. Here we show that a large fraction of exogenously induced persistent DDR markers is associated with telomeric DNA in cultured cells and mammalian tissues. In yeast, a chromosomal DNA double-strand break next to a telomeric sequence resists repair and impairs DNA ligase 4 recruitment. In mammalian cells, ectopic localization of telomeric factor TRF2 next to a double-strand break induces persistent DNA damage and DDR. Linear, but not circular, telomeric DNA or scrambled DNA induces a prolonged checkpoint in normal cells. In terminally differentiated tissues of old primates, DDR markers accumulate at telomeres that are not critically short. We propose that linear genomes are not uniformly reparable and that telomeric DNA tracts, if damaged, are irreparable and trigger persistent DDR and cellular senescence.