We employ interactions from chiral effective field theory and compute the binding energies and low-lying excitations of calcium isotopes with the coupled-cluster method. Effects of three-nucleon forces are included phenomenologically as in-medium two-nucleon interactions, and the coupling to the particle continuum is taken into account using a Berggren basis. The computed ground-state energies and the low-lying J(π) = 2+ states for the isotopes (42,48,50,52)Ca are in good agreement with data, and we predict the excitation energy of the first J(π) = 2+ state in (54)Ca at 1.9 MeV, displaying only a weak subshell closure. In the odd-mass nuclei (53,55,61)Ca we find that the positive parity states deviate strongly from the naive shell model.