Okay, people love telling this story, but the story is really a load of stuff coming out a horse's rear end.
Let's start with the track gauge. Standard gauge rail is 1435mm or 4'8½", but only about 55% of the world's track is standard gauge. Russia standardized on a 1520mm or 5' gauge, India uses a 5'6" gauge, the former British colonies in Africa use 3'6" gauge, and there's a few other gauges kicking about.
In the US itself, the South predominantly used a 5' gauge for all operations, until they shifted on May 31 and June 1, 1886 to the 4'9" gauge of the Pennsylvania Railroad (yes, the US shifted ~11k miles of track in just 36 hours). Notice how I said 4'9" there--the Pennsylvania Railroad, one of the largest railroads in US history, did not use 4'8½" gauge at the time but 4'9"; it only shifted to 4'8½" sometime in the early 20th century.
Go back in history, and the gauge proliferation was even worse, because there was little reason to standardize when railroads mostly didn't interconnect with each other. The notion that railroads adopted the distance between ruts on roads is largely bullshit because those distances weren't uniform. But if you're designing vehicles drawn by teams of horses, you're going to end up with a harness spacing of about 5', give or take a foot. The line that the Stephensons worked on used 4'8" (and was increased ½" for technical reasons) for its horse-drawn operations, so Stephenson built his locomotive to the dimension of the line, and used that dimension for later lines not directly connected to the original line. That seed of 4'8½" lines for interconnection eventually led to it winning out over alternative gauges, but the younger Stephenson is reported to have personally preferred a slightly broader gauge of perhaps 5'.
Now, the discussion of track gauge is perhaps interesting, but it's completely irrelevant to the sizing of SRBs for the Space Shuttle. The relevant metric there is what is known as the loading gauge--the maximum permissible envelope that can fit within tunnels and bridges (and station platforms). Track gauge is mostly irrelevant to loading gauge--unlike cars, where the body of the car sits between the wheels for the most part, train bodies sit above the wheels, and so the body can extend beyond the track gauge.
Loading gauge is far more variable than track gauge and is a more serious constraint on railroad operations than track gauge. Most European trains can't run in the UK, because the UK's loading gauge is absolutely tiny. Europe generally runs on a ~10'4" wide by ~13-14' high gauge. The US uses a very liberal gauge, with 10'8" by 14'6" the old norm but mostly replaced with a 20'2" gauge for double-stacked container freight.
A final point is that even the maximum size of a tunnel isn't determinative for parts. You can break parts into multiple pieces and assemble them after shipping. You can also ship oversize parts if you've got careful logistics planning. Ever seen oversize trucks on the road? There's similar games you can play with railroads; the oversized cargo wagons for freight are known as Schnabel cars, and you can do things such as shift your load so that it's off track center and get extra clearance in tunnels.
So to respond to the questions posed correctly:
> That's an exceedingly odd number. Why was that gauge used?
The ½" comes from a moderately arcane fix to an old coal railway to improve operation. Otherwise, 4'8½" is basically "the first successful railway used this."
> I see, but why did the English build them like that?
Because any number between ~4-6" is basically good enough, and the die in this one case just rolled on 4'8".
> Well, why did they use that gauge in England?
It was nowhere near universal, see prior answer. The answer given here is basically someone making stuff up. Remember, this is before interchangeable parts was deemed to be a workable concept, and standardization was not that precise.
> Okay! Why did their wagons use that odd wheel spacing?
As mentioned above, a team of horses two wide naturally needs you to 5±1' gauge. The answer given is pretty much crock. (Not going to bother with the next two questions, which are just doubling down on the crock).
> So, just what does this have to do with the exploration of space?
Nothing. The maximum dimensions of tunnels and bridges aren't as hard a constraint as it's made out to be, since there are several ways you can squeeze out extra space if you have to.
Just going to call out one bit that's particularly egregious.
> The tunnel is slightly wider than a railroad track
10'6" is not "slightly" wider than 4'8½", it's well over twice the size. If you've ever ridden a train in your life, you should realize that it is definitely far larger than just "slightly" wider than the railroad track.
P.S., for what it's worth, the SRBs on the Space Shuttle are 12'2" in diameter, according to Wikipedia. That doesn't fit in the normal loading gauge of even the US system.
Rail gauges are an interesting topic and there are a variety of gauges and many were experimented with over the years. Russia and Eastern Europe mostly runs on 1,520mm broad gauge. New Zealand and non-high-speed Japanese rails are 3'6". Wider gauges make trains more stable at high speed but require much larger corners, and vice versa.
Snopes rates it as partly true, and the author is a pedantic killjoy.
It is broadly true that for 2000 years European vehicles matched their roads and adapting existing vehicles to new technology led to using axles that were the same size. Do other gauges exist? Of course. Are the true elements of this worth repeating? Sure.
As an unrelated aside here’s Richard Feynman explaining why train axles don’t need differentials.
Honestly, rating it as partly true is giving it too much credit.
The only parts that are true are that a) the US adopted 4'8½" gauge largely from British standardization, and b) the 4'8½" gauge was chosen by extension of a pre-existing horse-drawn system. The moral of the story, however, is supposed to be that "the SRB was constrained by a Roman standardized wagon size by a long chain of events." And the first postulate is false before the chain can start (as far as I know, the Romans never standardized wagon sizes, not even informally--except insofar as you're going to end up picking roughly the same size, to within a foot or two, given similar constraints), while the last postulate is also false because track gauge does not determine loading gauge.
It also appears, judging from what I can find, that the loading gauge is actually smaller than the SRBs anyways. So even the loosest version of the final step of the chain appears to be false. So what's the elements worth repeating?
Modern computing analog: the first sentient AI to arrive in the Centauri system will contain CPU cores executing x86-64 instructions, a 64-bit extension to a 32-bit extension to a 16-bit segmented extension to an 8-bit teletype CPU.
echoes of an ancient (inaccurate) joke: "Windows 95 is a 32 bit extension and a graphical shell for a 16 bit patch to an 8 bit operating system originally coded for a 4 bit microprocessor, written by a 2 bit company that can't stand 1 bit of competition".
As far as memory addressing goes, 64 bits should be good enough until the exabyte RAM era, and there is no reason to switch the whole arquitecture before that.
Let's start with the track gauge. Standard gauge rail is 1435mm or 4'8½", but only about 55% of the world's track is standard gauge. Russia standardized on a 1520mm or 5' gauge, India uses a 5'6" gauge, the former British colonies in Africa use 3'6" gauge, and there's a few other gauges kicking about.
In the US itself, the South predominantly used a 5' gauge for all operations, until they shifted on May 31 and June 1, 1886 to the 4'9" gauge of the Pennsylvania Railroad (yes, the US shifted ~11k miles of track in just 36 hours). Notice how I said 4'9" there--the Pennsylvania Railroad, one of the largest railroads in US history, did not use 4'8½" gauge at the time but 4'9"; it only shifted to 4'8½" sometime in the early 20th century.
Go back in history, and the gauge proliferation was even worse, because there was little reason to standardize when railroads mostly didn't interconnect with each other. The notion that railroads adopted the distance between ruts on roads is largely bullshit because those distances weren't uniform. But if you're designing vehicles drawn by teams of horses, you're going to end up with a harness spacing of about 5', give or take a foot. The line that the Stephensons worked on used 4'8" (and was increased ½" for technical reasons) for its horse-drawn operations, so Stephenson built his locomotive to the dimension of the line, and used that dimension for later lines not directly connected to the original line. That seed of 4'8½" lines for interconnection eventually led to it winning out over alternative gauges, but the younger Stephenson is reported to have personally preferred a slightly broader gauge of perhaps 5'.
Now, the discussion of track gauge is perhaps interesting, but it's completely irrelevant to the sizing of SRBs for the Space Shuttle. The relevant metric there is what is known as the loading gauge--the maximum permissible envelope that can fit within tunnels and bridges (and station platforms). Track gauge is mostly irrelevant to loading gauge--unlike cars, where the body of the car sits between the wheels for the most part, train bodies sit above the wheels, and so the body can extend beyond the track gauge.
Loading gauge is far more variable than track gauge and is a more serious constraint on railroad operations than track gauge. Most European trains can't run in the UK, because the UK's loading gauge is absolutely tiny. Europe generally runs on a ~10'4" wide by ~13-14' high gauge. The US uses a very liberal gauge, with 10'8" by 14'6" the old norm but mostly replaced with a 20'2" gauge for double-stacked container freight.
A final point is that even the maximum size of a tunnel isn't determinative for parts. You can break parts into multiple pieces and assemble them after shipping. You can also ship oversize parts if you've got careful logistics planning. Ever seen oversize trucks on the road? There's similar games you can play with railroads; the oversized cargo wagons for freight are known as Schnabel cars, and you can do things such as shift your load so that it's off track center and get extra clearance in tunnels.
So to respond to the questions posed correctly:
> That's an exceedingly odd number. Why was that gauge used?
The ½" comes from a moderately arcane fix to an old coal railway to improve operation. Otherwise, 4'8½" is basically "the first successful railway used this."
> I see, but why did the English build them like that?
Because any number between ~4-6" is basically good enough, and the die in this one case just rolled on 4'8".
> Well, why did they use that gauge in England?
It was nowhere near universal, see prior answer. The answer given here is basically someone making stuff up. Remember, this is before interchangeable parts was deemed to be a workable concept, and standardization was not that precise.
> Okay! Why did their wagons use that odd wheel spacing?
As mentioned above, a team of horses two wide naturally needs you to 5±1' gauge. The answer given is pretty much crock. (Not going to bother with the next two questions, which are just doubling down on the crock).
> So, just what does this have to do with the exploration of space?
Nothing. The maximum dimensions of tunnels and bridges aren't as hard a constraint as it's made out to be, since there are several ways you can squeeze out extra space if you have to.
Just going to call out one bit that's particularly egregious.
> The tunnel is slightly wider than a railroad track
10'6" is not "slightly" wider than 4'8½", it's well over twice the size. If you've ever ridden a train in your life, you should realize that it is definitely far larger than just "slightly" wider than the railroad track.
P.S., for what it's worth, the SRBs on the Space Shuttle are 12'2" in diameter, according to Wikipedia. That doesn't fit in the normal loading gauge of even the US system.