  Concerned with elliptic operators with stationary random coefficients of
integrable correlations and bounded Lipschitz domains, arising from stochastic
homogenization theory, this paper is mainly devoted to studying
Calder\'on-Zygmund estimates. As an application, we obtain the homogenization
error in the sense of oscillation and fluctuation, respectively. These results
are optimal up to a quantity $O(\ln(1/\varepsilon))$, which is caused by the
quantified sublinearity of correctors in dimension two and the less smoothness
of the boundary. In this paper, we find a novel form of \emph{minimal radius},
which is proved to be a suitable tool for quantitative stochastic
homogenization on boundary value problems, when we adopt Gloria-Neukamm-Otto's
strategy originally inspired by the pioneering work of Naddaf and Spencer.
