That is, is there a city with multiple pro sports teams that all won their respective championships in the same year? If not, have there been any that came close?
For a while in 1928, New York held all three of the championships (the NBA wasn’t around yet). The Yankees won the '27 Series, the Giants won the NFL championship[sup]1[/sup] in 1927, and the Rangers won the 1928 Stanley Cup. The Yankees also won the '28 World Series, so until the Providence Steam Roller won the 1928 NFL, New York was king of the professional sports world.
[sup]1[/sup] By coming out on top of the league standings – the NFL championship game was first played in 1933.
I assume the OP only cares about the four “major” sports: Major League Baseball, the NFL, the NBA and the NHL.
There are MANY instances of cities winning championships in TWO of those major sports in the same year, So sorry, Bostonians, you’re far from unique. Just to give a FEW examples, the Pittsburgh Steelers and Pirates were champs in 1979, while the New Your Jets and Mets were champs in 1969.
NO town has won championships in all four sports in the same year. Dtilque already provided the one and only example I could find of a town with three of them.
The closest case recently? The Jets won the Super Bowl in January of 1969 and the MEts won the World Series in OCtober of 1969. Meanwhile, the Knicks won the NBA title for 1969-70 in the Spring of 1970… but by that time, the KC Chiefs had won the January 1970 Super Bowl.
This question was posted reletively recently but I cannot find a link.
In 1935 Detroit won the NHL, MLB and NFL championships.
1986, Boston Celtics won, Pats went to Super Bowl, Red Sox to World Series. Don’t know about Bruins. That’s a couple wins away from something 
If I’m reading the OP correctly, we’re not looking for a city that has won all of the nation’s major sports championships; we’re looking for a city that has won all for which it has teams. So if a city has NFL, MLB, and NHL teams, but no NBA, it would only need to win in football, baseball, and hockey to qualify.
Of course, there have been years when every pro team in Green Bay has won their championship, but the OP did specify more than one team. Perhaps we should be looking for cities with teams in only two of the major sports?
And I also note that nothing in the OP specifies the United States. Perhaps there’s a city in the British Isles with successful rugby, soccer, and cricket teams, for instance? I’m comfortable with counting whatever the major sports are in a particular region, rather than setting the same list for everyone.
The Mets won the World Series in 1986, and the Giants had a successful 1986 regular season, leading to a win in Super Bowl XXI in 1987. That’s more recent than the 1969 example given. In 1994, it wasn’t just one city, but one arena, that came very close: the New York Rangers won their first Stanley Cup in 54 years, and the Knicks made it to the NBA Finals, before losing to Houston.
We also have to clarify the issue of “city” vs. “metro area”. Would it suffice for the Oakland Athletics, the Raiders and the Golden State Warriors (we ARE indulging in fantasy here) to win their respective championships, or do the San Jose Sharks have to win the Stanley Cup, too? If that’s the case, I guess it counts if you substitute a Giants World Series or a 49ers Super Bowl for the A’s or the Raiders.
For that matter, does it count as some sort of lesser accomplishment if a city has two franchises in a sport, one of which wins the championship?