The first electrical-transport measurements of monatomic carbon chains are reported in this study. The chains were obtained by unraveling carbon atoms from graphene ribbons while an electrical current flowed through the ribbon and, successively, through the chain. The formation of the chains was accompanied by a characteristic drop in the electrical conductivity. The conductivity of the chains was much lower than previously predicted for ideal chains. First-principles calculations using both density functional and many-body perturbation theory show that strain in the chains has an increasing effect on the conductivity as the length of the chains increases. Indeed, carbon chains are always under varying nonzero strain that transforms their atomic structure from the cumulene to the polyyne configuration, thus inducing a tunable band gap. The modified electronic structure and the characteristics of the contact to the graphitic periphery explain the low conductivity of the locally constrained carbon chain.