This paper proposes a delayed sliding mode observer (DSMO) for the robust estimation of linear systems with bounded unknown inputs and measurement delays. In the DSMO, a static proportional minus delay output feedback loop is artificially incorporated, and an addictive switching delayed sliding mode injection is also introduced to achieve the convergence of estimation errors. The proposed DSMO permits both periodic and aperiodic measurements in output feedback loops, which improves the design flexibility with low-cost sensors. In simulation, an example of a magnetic levitation system is used to illustrate the merits of DSMOs.