quarterlife crisis
n. Feelings of confusion, anxiety, and self-doubt experienced by some people in their twenties, especially after completing their education.
This year an estimated 200,000 young people will embark on gap-year travels — not just school and college leavers leaping off the academic hamster wheel, but a growing number of restless twenty-somethings suffering from the 'quarterlife crisis'.
—Joanna Symons, “Mind the gap,” The Daily Telegraph, September 08, 2001
1998 (earliest)
Fox's Ally McBeal is making its retail inroads with a new line of 'loungewear' aimed at women in 'quarterlife crisis' and women with 'double standards to live up to.'
—Michael Stroud, “'Ally' hits Bloomingdale's,” Broadcasting & Cable, December 14, 1998
This phrase is a play on midlife crisis, which entered the language in 1965 thanks to Canadian psychologist Elliot Jaques, who wrote an article titled Death and the Mid-life crisis for the International Journal of Psycho Analysis. The quarterlife crisis variation was popularized by a book called Quarterlife Crisis: The Unique Challenges of Life in Your Twenties, by Alexandra Robbins and Abby Wilner, which was published in May, 2001. However, the authors of that book, numerous media assertions notwithstanding, didn't coin the phrase because it has been around for a few years (see the earliest citation).

In his 1991 book Generation X, Douglas Coupland coined a synonym: the mid-twenties breakdown, which he defined as
a period of mental collapse occurring in one's twenties, often caused by an inability to function outside of school or structured environments, coupled with a realisation of one's essential aloneness in the world. Often marks induction into the ritual of pharmaceutical usage.
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