This paper proposes a stochastic susceptible-acute-chronic-recovery (SICR) model for hepatitis C virus (HCV) transmission, in which environmental fluctuations in the transmission rate are modeled by a logarithmic Ornstein-Uhlenbeck process. This formulation preserves the positivity of parameters while capturing the mean-reverting stochasticity. A sufficient condition for disease extinction is established in terms of a stochastic threshold ${\mathcal{R}}_0^e$: if ${\mathcal{R}}_0^e<1$, then the infection becomes extinct almost surely. Conversely, when ${\mathcal{R}}_0^s>1$, the model admits at least one ergodic stationary distribution, which implies a long-term persistence of the disease. Furthermore, under the condition ${\mathcal{R}}_0>1$, an explicit probability density function for the stationary distribution near the quasi-endemic equilibrium is derived, thus providing a characterization of stationary fluctuations. Numerical simulations are conducted to validate the theoretical findings and to elucidate the influence of noise parameters on the disease dynamics. The results demonstrate that environmental fluctuations can sustain HCV transmission even when the deterministic reproduction number is below unity, thus underscoring the critical role of stochasticity in HCV transmission.