How is it possible that one eye can be both myopic and hypermetropic?

I wonder if you mean to ask "How can one eye be both nearsighted AND farsighted."

Unless one is talking about mixed (compound) astigmatism, you can't actually have a myopic (nearsighted) AND hyperopic (also called "hypermetropic") eye. But you can be nearsighted AND farsighted. There are TWO types of what people call farsightedness: hyperopia and presbyopia. Presbyopia is the age-related farsightedness that causes the slow deterioration of reading vision at near. Presbyopia, by itself, does not affect distance vision… but hyperopia can. Depending on the degree of hyperopia, both near and far vision can be blurred. The most common cause of distance vision problems, though, at least in people under 40, is myopia (nearsightedness),

Myopia is primarily associated with an eyeball that is "too big." The default focus in myopia is not off at a distance as it is in normally sighted people. Rather, it is "up close" depending on how myopic the eye is. A nearsighted person needs some kind of lens to help see at distance. The one good thing about myopia is that once you reach the age when presbyopia starts, a myopic person can just take off his or her glasses to see up close. This is because the up close distance is the natural, default focus of a myopic eye. But if you put back on your myopic correction (glasses or contact lenses) to see at a distance, you are then at the mercy of the up close blur from presbyopia.

_Written by J. Trevor Woodhams, M.D. - Chief of Surgery, Woodhams Eye Clinic