We consider simple scalar theories with quadratic terms that are nonlocal and Lorentz violating. Unlike similar Lorentz-invariant nonlocal theories that we have considered previously, the theories studied here are both ghost-free and unitary as formulated in Minkowski space. We explore the possibility that the scale of nonlocality could be low in a dark sector, where the stringent constraints on the violation of Lorentz invariance may be accommodated via the weak coupling to the standard model. We point out that long-range forces may originate from such a sector and be distinguishable from more conventional beyond-the-standard-model possibilities. We present a model in which a nonlocal, Lorentz-violating dark sector communicates with the standard model via a sector of heavy vectorlike fermions, a concrete framework in which phenomenological constraints and signals can be investigated.