The last dynasty in Major League Baseball started nearly a quarter-century ago, a run of dominance that defined a generation of the game’s history. In the 15 seasons between 1995 and 2009, the New York Yankees reached the playoffs 14 times, claimed seven American League pennants and won five World Series.

But as the 2019 season begins, the Yankees find themselves in a nearly unprecedented position. They last competed for a title 10 years ago—a blip for most teams, but an eternity for the most successful franchise in North...