Goebel was born on July 31, 1952, in Siegen, then West Germany. He began lessons on the violin at age 12 and attended the Cologne Conservatory, where he studied with Franzjosef Maier. He would round out his violin education by studying with Saschko Gawriloff at the Folkwangschule in Essen and Baroque violin with Marie Leonhardt and Eduard Melkus. In 1973, while studying musicology in Cologne, Goebel founded the legendary historical instrument ensemble Musica Antiqua Köln. Initially made up of other students at the university who were similarly interested in the music of the 17th and 18th centuries, Musica Antiqua would become one of the most influential early music ensembles in the world. Goebel was the violinist and director of the Musica Antiqua for the entirety of its existence. He focused his efforts with the group on unknown or seldomly performed composers, such as Johann David Heinichen and Jan Dismas Zelenka, as well as music by the masters, such as Bach and Telemann. The group's first recording was released in 1975 with the Kölner Vocal-Consort on an FSM Aulos album of music by Alessandro Grandi. From 1977, Goebel and Musica Antiqua recorded exclusively for Archiv Produktion, beginning with an album of recorder concertos by Francesco Mancini, Domenico Natale Sarri, Francesco Barbella, and Robert Valentine. This collaboration led to an extensive catalog and numerous awards, including several Deutscher Schallplattenpreis, the Grand Prix International du Disque, the Prix Caecilia, two Gramophone Awards, and several Grammy nominations. In 2007, Goebel announced his retirement from violin performance due to permanent injury issues with his hands, and with his retirement came the end of Musica Antiqua. The group's final recording was a 2005 album of Telemann flute quartets. Following the breakup of Musica Antiqua, Goebel focused his efforts full-time on conducting, leading both historical- and modern-instrument ensembles. Among the groups he's worked with are the Berlin Philharmonic, the Dresden Staatskapelle, all of the German Radio orchestras, and the Academy of Ancient Music. He is the principal guest conductor of the Bayerische Kammerphilharmonie, and in 2018, became the artistic director of the Berliner Barock Solisten. With this group, Goebel won an Opus Klassik award in 2018 for a recording of Bach's Brandenburg Concertos. Goebel focused a considerable amount of time in the late 2010s and early 2020s on a project he dubbed "Beethoven's World," which focuses on lesser-known Beethoven works and the music of some of his contemporaries. Included in this project is a series of albums, several of which appeared in 2020. Among these are a pair of releases with Mirijam Contzen: violin concertos of Franz Joseph Clement, and music by Salieri, Hummel, Vorisek. That year, Goebel also joined Benjamin Appl for an album of cantatas by the Bach family.
Goebel has taught historical performance since 2010 at the Mozarteum in Salzburg.