This paper analyzes the performance of a cooperative relaying non-orthogonal multiple access (NOMA) network over Fisher-Snedecor F composite fading channels. Specifically, one base station (BS) is assumed to communicate with two receiving mobile users with one acting also as a decode-and-forward (DF) relay. To highlight the achievable performance gains of the NOMA scheme, conventional relaying with orthogonal multiple access (OMA) is also analyzed. For the two systems under consideration, we derive novel exact closed-form expressions for the ergodic capacity along with the corresponding asymptotic representations. The derived expressions are then used to assess the influence of various system parameters, including the fading and shadowing parameters, on the performance of both the NOMA and OMA systems. Monte-Carlo simulation are provided throughout to verify the accuracy of our analysis. Results reveal that the NOMA system can considerably outperform the OMA approach when the power allocation factor is carefully selected. It is also shown that as the fading and/or shadowing parameters are increased, the ergodic capacity performance enhances.