Oceanic eddies play a major role in redistributing heat, salt, and tracers. Their three-dimensional structure has historically been studied from a dynamical perspective. Here, we investigate a more fundamental constraint: whether a balanced eddy can remain statically stable when embedded in a given oceanic environment. We prescribe idealized three-dimensional eddies from their vorticity structure and derive the associated density anomaly required for gradient-wind and hydrostatic balance. We then combine these anomalies with background density profiles representative of different regions and seasons to assess the resulting static stability. Our results show that static stability strongly depends on eddy polarity, intensity, vertical extent, and, for subsurface eddies, their position relative to the background pycnocline. Thin and intense cyclonic eddies are particularly prone to gravitational instability, whereas anticyclonic and thicker eddies are generally more stable. The same eddy structure can therefore be statically stable in one region or season but unstable in another, demonstrating that background stratification can constrain where particular eddy structures can exist and propagate. This process may also provide an additional pathway for eddy-induced vertical mixing through convective adjustment. We provide an open notebook that allows users to evaluate the static stability and three-dimensional density signature of prescribed eddies in arbitrary background stratifications.