We argue that the static dielectric constant of small (thin and/or narrow) semiconductor and insulator nanostructures depends strongly on the their environment. We do so by considering the electronic response simply reviewing, briefly but critically, the existing literature. Regarding the ionic response, in addition to reviewing the literature, we use a simple model to account for the confinement of optical phonons in thin films and show that the reduction of their density of states has a negligible effect on the dielectric constant, in contrast to some claims found in the literature. In general, we argue that in realistic structures, such as double-gated Si nanosheets, the use of the bulk dielectric constants for both the channel and the gate insulators, is justified.