Here's where Luke Skywalker is during the events of The Mandalorian. For the first live-action Star Wars TV show, creator Jon Favreau told a story that was mostly disconnected from the Skywalker Saga. Multiple references and Easter eggs for the larger Star Wars canon were included, but The Mandalorian season 1 mainly focused on a self-contained story. The series follows bounty hunter Din Djarin (Pedro Pascal) as he finds a baby of Yoda's species, and decides to protect it from the sinister forces who want to capture it.
The Mandalorian's ability to be separate from what the Star Wars movies and other content were doing was possible due to its setting. The show itself never outright confirms how long after the original trilogy these events take place, but Favreau previously said that it takes place five years after Star Wars: Return of the Jedi. This firmly puts The Mandalorian in 9 ABY (After the Battle of Yavin) in the Star Wars timeline, which is roughly 25 years before Star Wars: The Force Awakens. The first season does include some small, hidden time jumps, but nothing suggests The Mandalorian moves past 9 ABY.
Luke Skywalker has yet to appear or be referenced during The Mandalorian, but his whereabouts have been teased in the other canon materials. The show's 9 ABY setting means Luke already began training Leia to be a Jedi and went on the mission shown in Star Wars: Battlefront II. Those events took place in 5 ABY, and it isn't until 15 ABY that Luke begins training Ben Solo. It is in the middle of this gap that The Mandalorian takes place. Although the exact location of Luke in 9 ABY isn't confirmed, this period of his life was spent studying the Jedi so he could reform the Jedi Order. This required Luke to travel around the galaxy, finding pieces of Jedi history, and gaining a better understanding of what is required of the Jedi Order.
Due to Luke's clandestine activities, it's difficult to pinpoint where he is during The Mandalorian, as he could be just about anywhere. It is known that this is where he originally began to stay out of the spotlight and galactic events. Luke was frequently accompanied by Lor San Tekka during this time and visited locations with ties to the Jedi and Sith, such as Ahch-To and Mustafar. It was during these travels that Luke began collecting the books that Rey takes from him at the end of Star Wars: The Last Jedi.
Although Luke's Jedi missions are currently very separate from The Mandalorian's story, it might not be too difficult for that to change in the future. Din and Baby Yoda left to find the homeworld of Baby Yoda's species at the end of season 1, so it is possible that Luke could also journey to Yoda's planet a few years after his death. Meanwhile, the Darksaber is something that could draw the attention of Luke as well. The Darksaber is an ancient part of Jedi history, and Luke could begin searching for it if Moff Gideon's use of the weapon draws too much attention. After all, Luke could arguably be the rightful owner of the Darksaber at this point. With the inclusion of his father's former apprentice, Ahsoka Tano, in season 2, perhaps The Mandalorian is quietly building to a Luke Skywalker appearance - especially since it could work within canon.