Heisei Era refers to the Japanese history era in the original Fushigi Yuugi series. This is the time when Miaka and Yui went into the Universe of the Four Gods and summoning their respective Gods, Suzaku and Seiryuu, and some years later Mayo Sakaki was almost tricked into summoning a fake Suzaku. Additionally, the Priestess of Byakko is reunited with her lover Tatara in the afterlife, and her story is told to Miaka's brother Keisuke and his friend Tetsuya (offscreen in the manga, onscreen and by her grandson Toki in the anime series). Finally, if one counts the video games as at least alternate universes, it's also when other three young girls (Madoka Ootori, Misaki Himuro and Mariko Kobayashi) had their own adventures.
The Heisei era started in December 1989 and finished in April 2019, when Emperor Akihito abdicated the throne in favor of his son Crown Prince Naruhito, marking the end of the Heisei time and the start of the Reiwa Era. Akihito is the grandson of Emperor Yoshihito/Taisho (from when Genbu Kaiden, Byakko Ibun and the very first part of Byakko Senki took place) and son of Emperor Hirohito/Showa (from when Byakko Senki properly began). It's marked by both the rapid growth of Japanese economy as well as its military regrowth (as the JSDF alias Japan Self-Defense Dorces), but also by the problems coming from recession, several social and political crisis and changes through the decades, and many earthquakes.
The original Fushigi Yuugi manga began its run in 1992 and finished in 1996. It can be assumed that Miaka and Yui's adventures took place in a similar time frame, with Mayo's own story taking place some years later.