It is well known that defining a local refractive index for a metamaterial requires that the wavelength be large with respect to the scale of its microscopic structure (generally the period). However, the converse does not hold. There are simple structures, such as the infinite, perfectly conducting wire medium, that remain nonlocal for arbitrarily large wavelength-to-period ratios. In this work, we extend these results to the case of finite wire media with finite conductivity, using a two-scale renormalization approach. We show that the nonlocality of the homogeneous model is so extreme that the permittivity is geometry dependent. Its accuracy is tested and confirmed numerically via full-vector three-dimensional finite element calculations. Moreover, lossy finite wire media exhibit large absorption with small reflection, while their low fill factor allows considerable freedom to control other characteristics of the metamaterial, such as its mechanical, thermal, or chemical robustness.