One eye's worth. English has a small family of words built the same way: a is what a spoon holds, a handful is what a hand holds. An eyeful can be thought of as what an eye holds in one go. From there the word can point three ways: at the view that fills the eye, at the person or sight striking enough to be that filling, and at the real sand or water that fills the eye and makes it sting.
Wiktionary gives the formation simply as eye + -ful, the same suffix used in spoonful and handful to mean one container's worth.
來源資料把這個字的組成寫得很直接：eye 加上 -ful。-ful 就是 spoonful、handful 裡那個「一容器的量」的字尾，所以字面上就是「一眼睛的量」。這條線索不是後來的比喻，而是這個字本來的做法。
Wiktionary entry for eyeful (packet etymology_text: "From eye + -ful."; source_ref kaikki:3f6f69a190036ae4:1313349588:4110)
Three branches
眼睛裝滿的那一幕
最貼近核心畫面的一支：裝進眼睛的是景象。
noun · C1
A long, full look at something, so that you really take it in. It is almost always used in get an eyeful of something or give someone an eyeful, and the thing seen is usually surprising, shocking, or worth staring at.
看個夠的一眼
真的跑進眼睛裡的一份量
同樣是「一眼睛的量」，只是裝進來的不是畫面，而是實體的東西。
noun · C1
An amount of liquid, dust, sand or similar stuff that has got into someone's eye.
跑進眼睛的一堆東西
眼睛閉起來的那一小段
同樣的 -ful 玩法，但量的不是看進來的東西，而是眼睛閉著的時間。這是概念上的延伸，不是歷史來源的分支。
noun · C2
A short bit of sleep; forty winks. This is rare and informal, and it usually appears as an eyeful of sleep, with a verb like grab or catch.
打個小盹