In this paper we address the linear precoding and decoding design problem for a bidirectional orthogonal-frequency-division-multiplexing (OFDM) communication system, between two multiple-input-multiple-output (MIMO) full-duplex (FD) nodes. The effects of hardware distortions, leading to residual self-interference and inter-carrier leakage, are taken into account. In the first step, the operation of a FD MIMO OFDM transceiver is modeled under the impact of known hardware impairments. An alternating quadratic convex program (AltQCP) is then provided to obtain a minimum-mean-squared-error (MMSE) design for the defined system. The proposed design is then extended to maximize the system sum rate, applying the weighted-MMSE (WMMSE) method. The proposed AltQCP methods result in a monotonic improvement, leading to a necessary convergence to a stationary point. Finally, the performance of the defined system is evaluated under various system conditions, and in comparison to the other approaches in the literature. A significant gain is observed via the application of the proposed method as the hardware inaccuracy, and consequently inter-carrier leakage, increases.