It's not a record, but it's certainly unusual. And it's a very good sign.
Since 1903, the year of the first World Series, 25 other teams have begun a season with a run differential of at least plus-124 through their first 65 games (all historical data according to BaseballReference.com). Since 1940, there are nine other teams, and the list reads like a who's-who of some of the greatest teams in modern times.
The elite group consists of the 2001 Mariners, the 1998 Yankees, the 1998 Braves, the 1976 Reds, the 1976 Phillies, the 1974 Dodgers, the 1969 Orioles, the 1955 Dodgers and the 1944 Cardinals. Every one of those teams won at least 98 games, and all but the '55 Dodgers (who played only 153 games) won more than 100. Every one, of course, made the postseason.
Going back farther, the list includes legendary clubs like the 1927 and 1939 Yankees. The last 14 teams to start the season with a run like these A's finished in first place in their division or league, and they won an average of 105.5 games.
The highest run differential in a season's first 65 games, at least since 1903, belongs to the '39 Yankees at an almost unfathomable plus-211. They outscored their opponents by more than three runs per game for two months. That team went on to a 106-45 record and a world title behind some guys named DiMaggio, Dickey, Ruffing, and Gomez.
The highest since World War II was 147, posted by the 1969 Orioles and the 1955 Dodgers. The O's went 109-53 with Hall of Famers Frank Robinson, Brooks Robinson and Jim Palmer playing key roles. Oh, and their manager was a fellow named Weaver. They lost in the World Series to the "Miracle Mets," but their place as a great, great team is secure.
The 1955 Brooklyn club won it all with a group that included Jackie Robinson, Duke Snider, Gil Hodges, Don Newcombe and a kid named Koufax breaking in. It was the first Dodgers World Series title.
If we narrow our focus even further, to the last 35 years, only three teams have matched or equaled the A's start. And they are three of the greatest regular-season teams of that time span: the 116-win 2001 Mariners, the 114-win 1998 Yankees, and the 106-win 1998 Braves. That's it. That's the entire list of teams since 1980 that have outscored their opponents by as much as these A's over a season's first 65 games. Those are also three of the four teams in that span to win 106 or more regular-season games (the 1986 Mets are the other).
Only one of those three teams won the World Series, of course, but that's partly a testament to just how tough the modern postseason is (and how tough those dynasty-era Yankees teams were to beat). Besides, each of them won at least one postseason series.
In short, things look pretty sweet in Oakland. Their record tells you the A's are a very good team. Their run differential tells you we might be looking at something special, even historic. Either way, it will be fun to watch.