We investigate the nonstationary Navier-Stokes equations for an exterior domain $\Omega\subset \bm{R}^3$ in a solution class $L^s (0,T;L^q(\Omega))$ of very low regularity in space and time, satisfying Serrin's condition $\frac{2}{s} + \frac{3}{q} = 1$ but not necessarily any differentiability property. The weakest possible boundary conditions, beyond the usual trace theorems, are given by $u|_{\partial\Omega} = g \in L^s (0,T;W^{-1/q,q}(\partial\Omega))$ , and will be made precise in this paper. Moreover, we suppose the weakest possible divergence condition $k = \div u \in L^s(0,T;L^r(\Omega))$ , where $\frac{1}{3} + \frac{1}{q} = \frac{1}{r}$ .