We study a system of fermions in one spatial dimension with linearly confining interactions and short-range disorder. We focus on the zero temperature properties of this system, which we characterize using bosonization and the Gaussian variational method. We compute the static compressibility and ac conductivity, and thereby demonstrate that the system is incompressible, but exhibits gapless optical conductivity. This corresponds to a Mott-glass state, distinct from an Anderson and a fully gapped Mott insulator, arising due to the interplay of disorder and charge confinement. We argue that this Mott-glass phenomenology should persist to non-zero temperatures.
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