Because you must use a hyphen in something like “a half-day” or “a quarter-hour”. Otherwise you need to make it longer: half a day, a quarter of an hour.

Is it half-day or half-day?

A half-day is a day when you work only in the morning or in the afternoon, but not all day.

Is two and a half days hyphenated?

There is no need for hyphens if you're using the phrase as a noun: We'll be there in two and a half hours; two and a half hours is plenty of time. If you are using a phrase like that as a modifier, however, you'll need hyphens to hold it all together: a two-and-a-half-hour trip.