  We introduce a class of spin models with long-range interactions---in the
sense that they extend significantly beyond nearest neighbors---whose ground
states can be constructed analytically and have a simple matrix product state
representation. This enables the detailed study of ground state properties,
such as correlation functions and entanglement, in the thermodynamic limit. The
spin models presented here are closely related to lattice gases of strongly
interacting polar molecules or Rydberg atoms which feature an excluded volume
or blockade interaction. While entanglement is only present between spins that
are separated by no more than a blockade length, we show that non-classical
correlations can extend much further and analyze them through quantum discord.
We furthermore identify a set of seemingly critical points where the ground
state approaches a crystalline state with a filling fraction that is given by
the inverse of the blockade length. We analyze the scaling properties in the
vicinity of this parameter region and show that the correlation length
possesses a non-trivial dependence on the blockade length.
