(Nobody's answered this one on stackexchange after several days.)

My brother built a garage whose horizontal cross-section is a rectangle that measures $45$ feet by $30$ feet. To make sure the right angles were accurate, he measured the two diagonals of the rectangle to see that they were equal. In inches, $$ \sqrt{540^2+360^2} \approx 648.999229548\text{ inches} = 54\text{ feet}+1\text{ inch} - \text{less than $0.001$ inches}. $$ It's a bit odd to come within a thousandth of an inch when rounding to the nearest inch, but there's more: The rounding error is about $1/1298$ inches. But $1298$ is of course not exact; it's $$ 1298+\frac{1}{24073+\text{fraction}}. $$ Coming within $1/24074$ units when rounding to the nearest unit is even rarer.

$1298\cdot24073>31\text{ million}.$ At that rate you can win lotteries.

(I found that $540^2+360^2 + 1^2 = 649^2$.)

Is this merely a case of an archangel descending from the supernatural realm to have some fun messing with our brains, or is something remarkable happening? Or more prosaically, can something intelligent be said about this or is that the whole story?