Epsilon-near-zero (ENZ) materials, i.e., materials with a vanishing real part of the permittivity, have become an increasingly desirable platform for exploring linear and nonlinear optical phenomena in nanophotonic and on-chip environments. ENZ materials inherently enhance electric fields for properly chosen interaction scenarios, host extreme nonlinear optical effects, and lead to other intriguing phenomena. To date, studies in the optical domain have mainly focused on nanoscopically thin films of ENZ materials and their interaction with light and other nanostructured materials. Here, we experimentally and numerically explore the optical response of individual nanostructures milled into an ENZ material. For the study, we employ 3D structured light beams, allowing us to fully control polarization-dependent field enhancements enabled by a tailored illumination and a vanishing permittivity. Our studies provide insight between complex near-fields and the ENZ regime while showcasing the polarization-dependent controllability they feature. Such effects can form the basis for experimental realizations of extremely localized polarization-controlled refractive index changes, which can ultimately enable ultrafast switching processes at the level of individual nanostructures.