Warning! SPOILERS for The Book of Boba Fett episode 1
While The Book of Boba Fett features the titular bounty hunter as he seeks to become a Tatooine crime lord, other Star Wars characters from the original trilogy of films are alive and have a variety of new roles after Return of the Jedi as well. Although the era of this new Disney+ series is largely a period of peace, characters such as Luke Skywalker, Han Solo, and Leia Organa aren't done being heroes or leaders by any means. Likewise, Emperor Palpatine is secretly still alive as well, slowly regaining his dark power.
In The Book of Boba Fett episode 1, it's finally revealed how Fett survived the Sarlacc Pit following his supposed demise in Return of the Jedi. After being held hostage and later accepted by a tribe of Tusken Raiders, Boba Fett would retrieve his armor and join Din Djarin's rescue attempt of the child Grogu from an Imperial remnant, as seen in The Mandalorian. Now, Boba Fett has returned to Tatooine to become a crime lord after the late Jabba the Hutt, taking place approximately five years after the Battle of Endor and the destruction of the second Death Star.
During this same time period, many of the original trilogy characters are apart from one another such as Han Solo and Chewbacca being on different worlds, and the same goes for C-3PO and R2-D2. While it's unlikely Boba will cross paths with any of them in The Book of Boba Fett, it's still interesting to take a look at where each character is during the same time period.
Luke Skywalker
Being the original trilogy character most recently seen on screen, Luke Skywalker was featured in The Mandalorian's final episode for season 2, seeing the full-fledged Jedi Master agreeing to take and train the Force-sensitive child Grogu after helping to save him from the hands of the Imperial Moff Gideon. Past Star Wars projects - such as books, comics, and video games - have revealed that Luke is actively searching for ancient relics and writings of the old Jedi Order, hoping to train a new generation of Jedi now that the Empire and its Sith lords are (seemingly) no more. While Han Solo and Leia Organa's son Ben is about four years old at this time, he wouldn't be taken on as Luke's apprentice until he becomes 10.
Leia Organa
While also being a new mother to young Ben, Leia Organa is a very important senator of the New Republic, working to install a brand new democratic government in the wake of the Empire's defeat. As was revealed in The Mandalorian, starting up the New Republic is a long process, one that has only just begun to reach the Outer Rim territories such as Tatooine and Fett's new criminal enterprise. As such, it's unlikely Boba will face any kind of New Republic interference as The Book of Boba Fett continues. It's also important to note that Star Wars' Leia decided not to continue her Jedi training with her brother Luke shortly after the Battle of Endor, around five years before the new Star Wars series.
Han Solo
Struggling with fatherhood and the lack of a Force connection Leia holds with Ben, Han Solo can be perfectly defined during this era as restless. Having left his days of smuggling behind him while having no more war to fight in, Han Solo turns to starship racing to pass the time. He no longer has Chewbacca by his side either, seeing as how Solo helped Chewie liberate the Wookiee homeworld of Kashyyyk four years prior to The Book of Boba Fett. Han's struggles parenting Ben combined with Leia's busy schedule as a senator eventually lead to his return to smuggling in the years closer to The Force Awakens, especially after Luke takes Ben to be his apprentice.
Chewbacca
As previously mentioned, Chewbacca was aided by Han Solo in the liberation of Kashyyyk from Imperial control four years before the events of The Book of Boba Fett. Once they had achieved a Wookie slave revolution against the Imperial forces on the ground, they finally received support from a New Republic fleet led by Admiral Ackbar and Leia Organa who took out the Star Destroyers in orbit above the forest world. In the aftermath, Star Wars' Chewbacca reunited with the Wookiees of his homeworld, which is likely where he still is during Boba Fett's rise to become a crime lord. However, he would eventually rejoin Han when Solo decided to return to smuggling, helping him search for the Millennium Falcon, which had been stolen.
C-3PO & R2-D2
Despite having spent years together, the droid duo C-3PO and R2-D2 parted ways in the aftermath of the Battle of Endor. Five years later during the events of The Book of Boba Fett, Threepio is working as a protocol droid, fulfilling his primary function as an aid to Senator Leia Organa with the New Republic on Hosnian Prime, the new Galactic Senate's homeworld. Likewise, Artoo is with Luke Skywalker helping him locate old Jedi temples in search of artifacts and relics which could help him restart the Jedi Order. He was also present aboard Moff Gideon's light cruiser when Luke agreed to take Grogu.
Palpatine
Despite the large period of peace in the galaxy in the years after Return of the Jedi, Emperor Palpatine is not as defeated as the heroes of the Rebellion currently believe during this time period. While Darth Vader was redeemed by rejecting his master and destroying the Emperor's physical body on board the Death Star, Palpatine had contingency plans which were revealed in The Rise of Skywalker. Not only had Palpatine been building a new fleet of Star Destroyers on the Sith world of Exegol since the original trilogy (revealed in Marvel's current Darth Vader comic series) but he had imperfect clone bodies of himself constructed for his consciousness to reside in while he slowly regained his power.
While it's unknown whether or not Palpatine has already been reborn in a failing clone body during the same era as The Book of Boba Fett, his designs and master plan are no doubt in full swing. For example, the majority of the lingering Imperial remnants under Grand Admiral Sloane pulled back to the Unknown Regions after the Battle of Jakku, eventually reforming into the First Order under the rule of Supreme Leader Snoke, a puppet and prelude to the Emperor's return with his Final Order in The Rise of Skywalker.
The Book of Boba Fett releases new episodes Wednesdays on Disney+.