Yes they did! Their sorts were similar to Greek and Roman sorts, but things were a bit different because the empire had been Christianized. Roman blood sports like gladiatorial fights and mock hunts of wild animals were either legally prohibited or simply abandoned. Constantine made gladiator contests illegal in the 4th century, and athletic events that were pagan celebrations were also banned - so the ancient Olympics and other pagan Greek games were banned by emperor Theodosius in the 4th century as well. People could still practice the actual events of the games (running, discus, javelin, etc.), but the pagan festivities surrounding them were banned. Instead, festivals were organized around Christian holidays.

Everyone’s favourite sport in the early empire was chariot racing. It was an extremely popular sport in the ancient world and survived into the Byzantine Empire because it wasn't a blood sport, at least not intentionally - although it could get pretty violent. The hippodrome in Constantinople was right in the middle of the city, connected to the imperial palace and the Hagia Sophia cathedral. There were usually four teams, Reds, Greens, Blues, and Whites, and they all had their fans, like modern sports teams. The Blues and Greens were the biggest factions in Constantinople and even the emperors were fans - Justinian was a fan of the Blues. Sometimes the fans got into fights and riots, and in 532 the rioting got out of control. Much of the city burned down and thousands of people were killed (the “Nika riots”)

Chariot racing declined a bit after that but races and other games continued to be held in the hippodrome at least until the Fourth Crusade captured Constantinople. Races and other events in the hippodrome were linked to imperial inaugurations, victory celebrations, etc. Imagine if the US resident was inaugurated at an NFL football game! Or the Queen gave speeches before Premier League games...that’s sort of the same idea.

Later the association between the emperors and the hippodrome declined. The emperors moved to a new palace, although the hippodrome was still used for races, theatrical events and circuses (in the more modern sense, with animal tricks, acrobats, etc.)

"Byzantine mass amusements became less spectacular and less officially contrived in the eleventh and twelfth centuries. Roman horse racing had continued to be exceedingly popular during the first centuries of Byzantine history, even playing a part in state ceremonial. By the eleventh century, however, the circus spectacle was relegated to a minor role in Byzantine social life; its place was taken by the carnival. In contrast to the spectator sort of the circus, the carnival, with its masquerading, carousing, and buffoonery, allowed for the full participation of the common man." (Kazhdan and Epstein, pg. 82-83)

That was something like a modern carnival, in the religious sense like Mardi Gras. So, not a sport exactly, but it was the last time you could eat meat and your last chance to have fun before the more solemn season of Lent and Easter. The church and the emperors would also distribute food and other acts of charity, also in the hippodrome.

When the crusaders looted the city in 1204, they destroyed some of the hippodrome - there is a massive statue of a four-horse chariot in Venice, which they took from Constantinople after the crusade.

There were some other well-known sports in the Byzantine Empire but they were practiced by the emperors and aristocrats, not the common people. Polo, originally a Persian game, was especially popular. There were polo grounds in Constantinople and other cities. It wasn’t really polo exactly like the modern game - they called it “tzykanon” and it involved chasing a ball with a stick on horseback, so something polo-like, but maybe also with a net on the stick, like lacrosse.

The upper classes also enjoyed hunting and archery, and after increased contact with western Europe during the crusades, they also adopted western-style jousting tournaments. Those were all pretty dangerous and people could be killed, even emperors sometimes - John Komnenos for example died when he was accidentally injured by a poison arrow while hunting.

Sources:

Barbara Schrodt, “Sports of the Byzantine Empire”, Journal of Sport History, Vol. 8, No. 3 (1981), pp. 40-59 - not really a scholarly study by an exert, but a helpful summary of the toc

Alan Cameron, Circus Factions: Blues and Greens at Rome and Byzantium (Oxford, 1976)

Alexander P. Kazhdan, Ann Wharton Epstein, Change in Byzantine Culture in the Eleventh and Twelfth Centuries (University of California Press, 1985)

Cyril Mango, “Daily life in Byzantium,” in Jahrbuch der Österreichischen Byzantinistik 31 (1981), pg. 337-53

Phédon Koukoulès, Vie et civilisation Byzantine - this is the French translation of a work that’s originally in Greek, so I’m not sure how helpful that would be, and I haven't been able to find it myself, but apparently he talks about sports in volume 3.