I know it is not possible to prove the phrase but we can add or remove plausibility to the theory about this having to do with belts of ammunition. I will add what I can and you all can do with the information as you like.
100 rounds of 50 cal ammunition is 8 feet long. Yes the belt will stretch and grow a little due to the clearance in the links but it is not very much at all.


This works out to 337 rounds in 27 feet (pretty much agrees with what Red Baaron came up with). The ANM-2 50 cal aircraft gun fires between 750 and 850 rounds per minute or 12 to 14 rounds per second. The whole 9 yards means holding the trigger down for 24 to 27 seconds which sure is a long time and I imagine would cook the barrel.
I would find it hard to envision any gun run either strafing or air to air where you might have the opportunity to fire for such an extended period. Ground gun firing seems more likely. Some of those haunting scenes from the German side of Normandy Beach in Saving Private Ryan come to mind.
Here is a start for ammo capacities for various aircraft. Please add to the list as you like as these were the only manuals close by. These numbers are general in nature and your actual capacities may vary. Some numbers are from the internet so watch out

Fighters
P-40E 281 rounds per gun
P-51B&C 280 outboard 350 inboard guns
P-51D 400 inboard and 270 middle and outboard guns
P-47D 260 to 425
F6F-3 400 RPG
F4U 390 RPG
Turrets
Bendix upper (B-25) 400 rounds per gun
Bendix Chin (B-17G) 365 rounds per gun
Consolidated tail (B-24) 400 rpg
Sperry A-1, A-1A upper (B-17 E,F,G) 375 RPG
Sperry A-1B upper and A-1C (B-17G) 750 RPG
Sperry A-2, A-2A, A-2B, A-13 Ball turret (B-17, B-24) 500 RPG
Martin upper (B-24, A-20, PV-1, PV-2, PB4Y-2, B-26) 400 RPG

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