The hardware impairments of transceiver nodes that generate signal distortions degrade the reliability performance of communication systems. This paper investigates the impact of hardware imperfections on the outage performance of hybrid satellite-terrestrial relay systems (HSTRS), where a geosynchronous earth orbit (GEO) satellite transmits its data to the terrestrial destination with the help of decode-and-forward (DF) terrestrial relays. We consider that all network nodes are subjected to hardware impairments and propose a multi-relay selection (MRS) scheme to enhance the outage performance of the HSTRS. The traditional single-relay selection (SRS) scheme is also considered for comparison purposes. We derive closed-form outage probability expressions for both the SRS and MRS schemes. Moreover, an asymptotic outage probability analysis is carried for both the SRS and MRS schemes in the high signal-to-noise-ratio (SNR) region. The simulation results show that the outage performance of the HSTRS can be enhanced under the MRS scheme with a certain level of hardware impairments at the HSTRS. Compared to the condition of ideal hardware, the outage probability of HSTRS increases for both the SRS and MRS schemes if the hardware is not ideal. When the amplify-and-forward (AF) protocol is employed at relays, the outage performance of HSTRS degrades compared to that of DF for both the SRS and MRS schemes.