Sliding mode based feedback control has long been recognized as a powerful, yet easy-to-implement, control method to counteract non-vanishing external disturbances and unmodelled dynamics. Recently, research attention has focused on the development of sliding mode feedback control methods for various classes of infinite-dimensional systems. However, the existing methods are based on the assumption that distributed sensing and actuation is available, which significantly restricts their applicability to distributed process control applications. In this work, a sliding mode output feedback control method is developed for a class of linear infinite-dimensional systems with finite-dimensional unstable part using finite-dimensional sensing and actuation. Modal decomposition is initially used to decompose the original infinite-dimensional system into an interconnection of a finite-dimensional ( possibly unstable) system and an infinite-dimensional stable system. Then, a sliding mode-based stabilizing state feedback controller is constructed on the basis of the finite-dimensional system. Subsequently, an infinite-dimensional Luenberger state observer, which utilizes a finite-number of measurements, is constructed to provide estimates of the state of the infinite-dimensional system. Finally, an output feedback controller design is completed by coupling the infinite-dimensional Luenberger state observer and the sliding mode-based state feedback controller. Implementation, performance and robustness issues of the sliding-mode output feedback controller are illustrated in a simulation study of a distributed parameter system governed by the linearization around the spatially-uniform steady-state solution of the Kuramoto-Sivashinsky partial differential equation with periodic boundary conditions.