We study optimal procedures for estimating a linear functional based on observational data. In many problems of this kind, a widely used assumption is strict overlap, i.e., uniform boundedness of the importance ratio, which measures how well the observational data covers the directions of interest. When it is violated, the classical semi-parametric efficiency bound can easily become infinite, so that the instance-optimal risk depends on the function class used to model the regression function. For any convex and symmetric function class $\mathcal{F}$, we derive a non-asymptotic local minimax bound on the mean-squared error in estimating a broad class of linear functionals. This lower bound refines the classical semi-parametric one, and makes connections to moduli of continuity in functional estimation. When $\mathcal{F}$ is a reproducing kernel Hilbert space, we prove that this lower bound can be achieved up to a constant factor by analyzing a computationally simple regression estimator. We apply our general results to various families of examples, thereby uncovering a spectrum of rates that interpolate between the classical theories of semi-parametric efficiency (with $\sqrt{n}$-consistency) and the slower minimax rates associated with non-parametric function estimation.