  We prove the existence and the uniqueness of a local maximal solution to an
$H^1$-critical stochastic wave equation with multiplicative noise on a smooth
bounded domain $\mathcal{D} \subset \mathbb{R}^2$ with exponential
nonlinearity. First, we derive the appropriate deterministic and stochastic
Strichartz inequalities in suitable spaces and, then use them in arguments
based on fixed point method to show the local well-posedness result. We also
present an explosion result for the constructed unique local maximal solution.
