We prove that, at arbitrary positive temperature, every infinite-volume local limit point of the two-dimensional one-component plasma (2DOCP, also known as Coulomb or log-gas, or jellium) satisfies a system of Dobrushin-Lanford-Ruelle (DLR) equations - in particular, we explain how to rigorously make sense of those despite the long-range interaction. We also show number-rigidity and translation-invariance of the limiting processes. This extends results known for the infinite Ginibre ensemble. The proofs combine recent results on finite 2DOCP's and classical infinite-volume techniques.