When Does “Cowboy Bebop” (1998) Take Place?
"Kaubōi Bibappu" (released to English-speaking audiences as "Cowboy Bebop") is a science fiction animated television series showran by Shinichirō Watanabe that released April 3rd, 1998. So when is "Cowboy Bebop" set?
It takes place in the year:
We know this because in episode 21 "Boogie Woogie Feng-Shuei" (1999), Jet’s old friend Pao's gravestone says "2021 ~ 2071" and Pao died "3 days ago." Unless it’s happening on January 3rd, 2072 AD, the episode must be in 2071 AD.

2071 AD
We know this because in episode 21 "Boogie Woogie Feng-Shuei" (1999), Jet’s old friend Pao's gravestone says "2021 ~ 2071" and Pao died "3 days ago." Unless it’s happening on January 3rd, 2072 AD, the episode must be in 2071 AD.
More specifically, the same episode includes a print out of an e-mail Jet receives from his recently-deceased friend just before his death and it reads "RECEIVED/AUG.18.3:16 PM" in the corner. So it's probably August of 2071 AD during episode 21 "Boogie Woogie Feng-Shuei".
Additionally, episode 15 "My Funny Valentine" reveals the personal backstory of Faye Valentine and gives key background to the world of the show. A flashback in this episode is clearly dated (a rare example) to the year 2068 AD.
Multiple other episodes have quick mysterious flashbacks that are mostly unplaceable.

