When Does “Hyperion” (1989) Take Place?

"Hyperion" is a science fiction novel written by Dan Simmons and released May 26th, 1989. 

It takes place in the year: 

2764 AD 

We know this because Rachel Weintraub recorded a message to herself that began: 

"I'm recording this on the twelfth day of Tenmonth, year 457 of the Hegira, A.D. 2739 old reckoning. Yes, I know that's half a standard year from the last thing you remember." 

Rachel was 25 years old at the time of the accident that causes her to age backwards. Roughly 24 years and six months have passed because her father states that she is the equivalent of only seven weeks old at the present time of the novel. 

The journals of Father Paul Duré (read aloud by Father Lenar Hoyt) state: 

"I am somewhat at a loss as to how to date my new journal. By the monastic calendar on Pacem, it is the seventeenth day of Thomasmonth in the Year of Our Lord 2732. By Hegemony Standard, it is October 12, 589 P.C. By Hyperion reckoning, or so I am told by the wizened little clerk in the old hotel where I am staying, it is the twenty-third day of Lycius (the last of their seven forty-day months), either 426 A.D.C. (after dropship crash!) or the hundred and twenty-eighth year of the reign of Sad King Billy, who has not reigned for at least a hundred of those years." 

32 years passed since Duré wrote those words and 25 years have passed since Rachel's message to herself so, in these various calendars, 2764 AD is 621 PC, 458 ADC, and year 482 of the Hegira. 

It's important to remember that since the novel is structured as a series of personal stories told to a group [to intentionally parallel Geoffrey Chaucer's "The Canterbury Tales" (1400)], each of these flashback tales take place across a wide span of previous years.