I didn't know, so I told him I'd try to find out. I asked Larry Kass, the director of communications at Heaven Hill. Larry asked the folks at Independent Stave, and also asked the guy I should have gone directly to: Mike Veach, the closest thing bourbon has to an over-arching 'corporate memory.' Mike's an avid researcher, and a tireless advocate of funding for a bourbon archive and historical center. Here's his answer, sent through Larry:
As far as the barrels are concerned, this is what I have always heard: The barrels used to be a standard 48 gallons and that is the size the warehouse ricks was designed for storing. During the Second World War wood became scarce for cooperage and the decision was to increase the gallon size of the barrels to save wood and space in the warehouses - and to save cost as well but that was not the main point at the time.
The size of 53 gallons was the largest that they could make to fit in the standard ricks without making the structure of the barrel weak causing leaks. Now this information came from an article that was based upon second hand sources, but it does seem the most logical explanation. I was told by some warehouse workers that they thought if they made it larger, it would also be too hard to handle while rolling. They said the old 48 gallon barrels were much easier to handle, so the increased size made the barrels more difficult to roll. That could also have played a part in the size of 53 gallons.
So there you go. Makes sense. Thanks to Mike, to Larry, and to Tom for asking an interesting question.