The events of The Mandalorian don't take place long after Return of the Jedi, yet it doesn't feature any familiar Star Wars characters, but they still have stories going on in different places in the galaxy. More than 30 years after the original trilogy concluded in real life, Lucasfilm's first live-action Star Wars TV show picks up just a few years later. In the Star Wars canon, the events of the original trilogy concluded in 4 ABY (After Battle of Yavin). This meant that nearly two decades passed between Star Wars: Return of the Jedi and the returns of Luke Skywalker, Leia Organa, and Han Solo in Star Wars: The Force Awakens.
The sequel trilogy might be over after Star Wars: The Rise of Skywalker, but The Mandalorian continues to be the main attraction for the galaxy far, far away. The first season of Jon Favreau's Disney+ show primarily followed Din Djarin (Pedro Pascal) after he finds The Child aka Baby Yoda. Although the adventures of the two characters include plenty of familiar alien species, droid types, and planets, known characters from the larger Star Wars canon don't appear, including the original trilogy stars.
Fans have mostly been happy with The Mandalorian's decision to not include any familiar characters so far, but that appears to be changing with season 2. The influx of characters from across the Star Wars canon is mostly expected to include characters from animation and comics/books making live-action debuts. As of this time, the main characters from the original trilogy are not expected to appear. But, thanks to Disney's emphasis on fleshing out the timeline through various canon materials, there is a great deal of information known about what they're up to during The Mandalorian.
Luke Skywalker
Luke Skywalker's story around the events of The Mandalorian sees him continuing his quest to learn more about the Jedi Order. Luke spent this time searching the galaxy for Jedi and Sith artifacts, which lead him to a mission on Pillio to try and find Emperor Palpatine's Observatory. It is there that he finds the compass Luke has in his possession in Star Wars: The Last Jedi. Luke also began training Leia to use the Force a few years before The Mandalorian. After Leia stopped training, Luke turned his focus back towards rebuilding the Jedi Order.
This resulted in Luke largely disappearing from the view of the galaxy, as he traveled to multiple planets, including the Sith homeworld of Moraband. These adventures led him to meet Lor San Tekka and learn about the Jedi Masters who trained his father. He also gathered many of the books Rey uses in Star Wars: The Rise of Skywalker during this time. It's only after all of this, and a few years after The Mandalorian is currently set, that Luke begins training a new generation of Jedi.
Leia Organa
Leia Organa is in a brand new phase of her life during the events of The Mandalorian. On the day the Empire was finally defeated in 5 ABY, Leia gave birth to her first and only child, Ben Solo. By the time The Mandalorian takes place, Ben is just four years old and surely a handful for Leia. She has already decided by now not to pursue harnessing her Force abilities after having a vision that Ben would die if she continued down the path to becoming a Jedi. This decision was made even after building her own lightsaber and training with Luke.
Leia's duties as a mother are her priority during The Mandalorian, but she is also just beginning to think about what she can do politically to help the galaxy. She is no stranger to the political realm after being raised by Bail Organa, and Leia eventually has a successful career as a leader in the New Republic. Leia is later named the senator for the Alderaan sector of the galaxy, but that doesn't happen for a few years after the current story of the Disney+ series.
Han Solo
Han Solo's story in the sequel trilogy showed him returning to his roots as a smuggler, but he is in a very different position during The Mandalorian. According to canon materials, Han has spent the last few years married to Leia and raising their son Ben. His new life hasn't completely kept him out of the pilot's seat, though, as Han does still fly the Millennium Falcon from time to time. Han's exact whereabouts during The Mandalorian aren't clear, but it is known that he loses that Falcon to Gannis Duccain sometime around here.
Lando Calrissian
Lando Calrissian's life after the original trilogy is busy in the lead up to The Mandalorian. He has spent the last few years serving the New Republic. He assumes more responsibility as the days of the Empire's reign draw to a close. Lando worked with Iden Versio and Del Meeko during this time, which predates The Mandalorian by a few years, to prevent Operation: Cinder. Lando also played a role in liberating Cloud City and has a bounty on his head for the role he played in blowing up the second Death Star. In 7 ABY, Lando went on a mission with Han and Chewbacca to save Cloud City by returning a Phylanx Redux Transmitter stolen by Han and Sana Starros several years ago.
Chewbacca
Chewbacca is retired from his days of saving the galaxy and smuggling when The Mandalorian takes place. He returned to his homeworld of Kashyyyk to liberate the planet from what remained of the Empire. In order to achieve this victory, Chewbacca worked with Han and several other mercenaries to free his family from the Empire's grasp. Chewbacca decided to stay on Kashyyyk following this victory instead of returning to work with the New Republic, although he would later rejoin Han to become smugglers again.
Boba Fett?
Even though it appeared Boba Fett died during Star Wars: Return of the Jedi, it is expected that The Mandalorian season 2 will confirm otherwise. There have already been hints in Star Wars canon that Boba Fett managed to survive his fall into the sarlacc living inside the Great Pit of Carkoon. It isn't yet known what Boba has done in the five years since his near-death experience, but it appears The Mandalorian season 2 will change that.