The intrinsic strength of freestanding carbon atomic chains was measured by in situ high-field mechanical testing of carbon atomic chains carried out inside a field-ion microscope. The determined breaking field strength corresponds to a tensile strength of carbon atomic chains at 5 K equal to 245 GPa. Carbon atomic chains also show exceptionally high-evaporation stability in electric fields up to 259 V/nm. The tensile strength of linear carbon chains significantly exceeds the tensile strength of known two-dimensional and three-dimensional carbon materials, including carbon nanotubes and graphene. Our results can be considered as direct experimental evidence for Pauling's prediction of bond stiffening with reduced atomic bond order.