Ink needs to get between the conjunctiva and the sclera, which are labeled 1 and 2 respectively above. The sclera is the actual white bit, and the conjunctiva is a thin, clear layer directly atop. In the body art community, there's basically just one method of tattooing: injecting small volumes of dye into the space between the two layers, and allowing that ink to spread in patches. It only requires a few needle pricks—unlike a traditional tattoo, where the tip has to repeatedly pierce the skin. Technically it would be possible to do that to an eyeball, but you'd mostly end up with a lot of holes for the dye to leak out of. Sorry for that mental image.