Well, no wonder they stunk! Frank Thomas hadn’t even been born yet! 
Actually, their hitting, while not tops, was not too bad for a last-place team. Their runs/game were comparable to the Cubs, and better than the Astros, the two teams that topped them in the standings. (Though it obviously helped that they played in the Polo Grounds).
What really made them stink was their defense – 210 errors, 19 more than the second worst team (the Dodgers, who had the pitching to overcome it) and 37 more than the third worst – and their pitching (5.89 runs/game, 0.79 runs more than the second worst NL team).
Age wasn’t that much of a factor. Though the stuffed the roster with old Dodgers and Giants, only three of their regulars were over 30, and that included Richie Ashburn and Frank Thomas (their two best players). Charlie Neal, the third, also had a decent year. The rest of the old guys (other than Gene Woodling) appeared in less than a third of their games.
The 1963 Mets were even worse, even though the lost only 111: nearly a half run per game scored (and still in the Polo Grounds), with the errors and runs given up a game about the same as the year before. A .219 team batting average (it was .246 the year before).
You know, we’ve forgotten the Chicago/St. Louis/Arizona Cardinals. Their last championship was in 1947, and they’ve only won one playoff game since then (I’m not counting their victory in the NFL 3rd Place Game). I would guess their franchise has more losses than any other NFL team (it dates back to 1920, and they once lost 29 games in a row).
Actually, the graphic featured the six franchises (including the Astros, who subsequently won the game and thus advanced to the World Series for the first time) that had never made it to the Fall Classic. Besides the expansion Senators that became the Rangers, the remaining “virgin” clubs are the Montreal Expos/Washington Nationals, Seattle Mariners, Colorado Rockies, and Tampa Bay Devil Rays.
The Cubs last won the Series in 1908, and last appeared in '45 – both current records for futility. Here is a list of all current major-league teams, the year they last won the Series, and the year they last made it:
Arizona Diamondbacks – 2001, 2001
Atlanta Braves – 1995, 1999
Baltimore Orioles – 1983, 1983
Boston Red Sox – 2004, 2004
Chicago Cubs – 1908, 1945
Chicago White Sox – 1917, 2005 (had been 1959)
Cincinnati Reds – 1990, 1990
Cleveland Indians – 1948, 1997
Colorado Rockies – never, never (franchise began in 1993)
Detroit Tigers – 1984, 1984
Florida Marlins – 2003, 2003
Houston Astros – never, 2005 (first appearance since inception in 1962)
Kansas City Royals – 1985, 1985
Los Angeles Angels of Anaheim/Anaheim Angels/California Angels – 2002, 2002
Los Angeles Dodgers – 1988, 1988
Milwaukee Brewers – never, 1982 (began as 1969 Seattle Pilots)
Minnesota Twins – 1991, 1991
New York Mets – 1986, 2000
New York Yankees – 2000, 2003
Oakland Athletics – 1989, 1990
Philadelphia Phillies – 1980, 1993
Pittsburgh Pirates – 1979, 1979
St. Louis Cardinals – 1982, 2004
San Diego Padres – never, 1998 (first season was 1969)
San Francisco Giants – 1954 (as New York Giants), 2002
Seattle Mariners – never, never (began play in 1977)
Tampa Bay Devil Rays – never, never (joined American League for 1998 season)
Texas Rangers – never, never (franchise began as 1961 Washington Senators)
Toronto Blue Jays – 1993, 1993
Washington Nationals – never, never (debuted as 1969 Montreal Expos)
Lots more Series information here.
Easy to remember with this song lyric:
You know, the law of averages says anything will happen that can (that’s what it says),
but the last time the Cubs won the National League pennant was the year we dropped the bomb on Japan.
Neither do I; I was just flirting.
Actually, one that I probably should have mentioned in the OP was the New Zealand cricket team, who in 1950-something were bowled out for 26 runs in an innings of a test match vs England.
It’s a record that will probably stand the test of time - a typical score will usually be well in excess of 350; and anything under 200 is considered pretty disappointing. If you Americans imagine an NBA team scoring around six points total in a single game you might get an idea of just how embarrassingly bad this was.