Though this word is extremely rare, the situation it refers to is still very much with us, as it refers to one of the corrections that form part of the Gregorian calendar we all use. We need not worry that it will affect us anytime soon, however: the next metemptosis will occur in 2100.
The problem for calendar creators is that the number of days in a year doesn’t exactly fit the length of the year — there’s about a quarter of a day over. The older Julian calendar from Roman times had a simple way to deal with this: it just added an extra leap day every four years to make up the numbers and get the calendar back into sync with the year.
Unfortunately, over the next 1500 years it slowly became clear this wasn’t good enough. In fact, the number of days in the year is very slightly less than 365¼, so the calendar was slowly gaining on the seasons. In the sixteenth century, advisors to Pope Gregory XIII told him that it was necessary to leave out some days to get things back in step and to change the calendar to omit three leap years in every 400 years. To do this, they suggested that century years should only be leap years if they were divisible by 400. So 2000 was a leap year, but 2100 won’t be.
Astronomers created metemptosis in the early eighteenth century for this second step, from Greek meta-, after, em-, in, and ptosis, a falling. It came into the language when the need to reform the British calendar was becoming very urgent (Catholic countries had implemented it in 1582; Britain only did so in 1752, though we weren’t the last by any means).
To be complete, there is proemptosis for the opposite process, adding a day to the calendar, in this case to keep it in step with the moon. For reasons I have no intention of trying to explain, one of these will not be needed until 4200.