From a previous question a portable hole can hold 282 cubic feet of stuff.
A barrel can hold 4 cubic feet of food from the description but that is only the interior. Would it be reasonable to say that a barrel itself takes up 4 cubic feet in the portable hole? That would mean 70 barrels could fit in the portable hole.
This calculation is a bit more complicated than just dividing the portable hole’s volume by the barrel’s volume since they will not fit perfectly and take up all the room, so there will be gaps in between of them. I’ve done some calculations and arrived at the conclusion that you could fit 21 barrels into the portable hole.
First, I looked up the dimensions of a barrel. There were lots of different types of barrels used throughout the ages but the general consensus is that a 40 gallon whiskey barrel would have a height of 88 cm and a diameter of 60 cm at its widest (roughly in the middle height-wise). I am going to treat the barrel as a perfect cylinder here, using its widest measurement as the diameter to simplify things since it doesn’t really make any difference in practice.
A portable hole, as noted in its description, is a cylinder with a height of 10 feet and diameter of 6 feet, which translates to 304 cm and 182 cm, respectively.
If we’re going to store the barrels in layers inside the portable hole, with the barrels standing upright, as it’s the most efficient method, we will be able to fit in 3 layers of barrels within the hole. This is calculated by dividing the hole’s height by the barrel’s height:
$$ \frac{304}{88} = 3.455 $$
That gives us 3 layers and about 40 cm of empty space at the top into which it’s impossible to fit in another barrel, even when putting it on its side (60 cm diameter so it would stick out by about 20 cm).
To calculate the maximum amount of barrels that would fit in each layer we need to basically solve the problem of smaller circles inside a larger circle, I have used this caclulator here for that: Smaller Circles within a Larger Circle - Calculator.
Putting in a portable hole’s diameter as the large circle and a barrel’s as the small one, the result it produces is 7. It would look like this viewed from the top:
So to put this together, 7 barrels in layer × 3 layers gives us 21 barrels that would fit into a portable hole.
A typical M1911 Barrel is roughly a 5 inch long cylinder of a diameter of 0.6 inch. That's 1.4 cubic inches. Using a form factor of 0.732 as we know from stacking coins, there's one barrel per 1.93 cubic inch. Or in other words: 252314 complete barrels, but no other parts. This method only works for tiny objects compared to the room volume, because these tiny items stack almost like coinage.
While the form factor would tell us, that there should be space for about 51 barrels in the hole if that hole had an arbitrary shape, the outer dimensions become problematic: Our hole has a fixed shape. 6 feet in diameter, 10 feet high. For the best packing, we assume 7 cylindrical barrels of 2 feet in diameter - or 1-foot radius - in a layer.
Assuming that the barrels are perfectly cylindrical, each barrel would need to be 1 foot 3 inches tall to fit the 4 cubic feet volume (\$\frac V \pi=h\$). Let's round that to 1.5 feet to account for at least some shell that holds these contents.
That means we could have 6 complete layers of 7 barrels or 42 barrels. Or rather, Jerry-cans, as we only accounted for a rather small shell here.
Now, let's assume the barrels have a 1-inch wood frame, and a 1 inch wood lid. That lid should be at least half an inch inch indented from the top and bottom of the barrel jacket's end. It needs to contain 4 cubic feet of volume or 113.27 liters. Since water has a density of pretty much 1, that's 113.27 kilos of water in that barrel. This happens pretty perfectly with the lid being between 0.9 and 1.9 inches from the top edge, when the barrel is 0.8 feet radius at top and bottom of the jacket. And we demand a round number of feet for the height, so math gets easier. After some experimentation, the measurements in millimeters read like this (sorry, my CAD only outputs mm, it allows me to put in variables in inch though):
This results in 2-feet tall barrels of 2 feet maximum outer diameter while the top and bottom of the barrel curve in to have a 1.6 feet diameter at the ends. Perfect!
Stacked as before, we get a perfect fit of 5 layers of 7 barrels each, for exactly 35 barrels in total.
40 gallons are 151.42 liters of water. Using the same basic barrel geometry as before, our 4 gallon barrels need to be 2.7 feet tall to get that (and a tiny bit) of volume.
We can still store 7 per layer, but we only manage to stack \$\frac {10}{2.7}=3,7\$ into the hole, so 3 complete layers for \$3*7=21\$ barrels
If you have two slightly ovaloid barrels or a pair of longer but slimmer barrels, you might manage to get those in as extra: on their flat side on top of the stack.
A proper Jerrycan is of the following dimensions: 165x345x470 mm. Stacking those in a portable hole gives this pattern of 38 cans per layer:
6 complete Layers fit into the hole, so 6*28=168 canisters filled with 3360 liters of water or fuel. Or 887,6181 gallons for the imperial types - the equivalent of a little over 22 barrels of 40 gallons. But the standing Jerry cans still leave just enough space above for a top layer of 10 extra cans laying on the side:
That's 200 more liters, for a total of 3560 liters or 940.4525 gallons. The equivalent of 23.5 barrels.
Wait, there's even a better stacking with 12 extra lying cans, for 40 more liters, so a total of 180 times the 20-liter canister. 3600 liters stacked in German engineering!
The barrels will not fill that space fully, as they are more or less cylindric in form. If they were ideal cylinders, you could apply circle packing in a circle as long as the height of the barrels fits in. We'll start from there.
The portable hole creates a cylindric space 10 feet deep and 6 across:
It unfolds into a circular sheet 6 feet in diameter. (…) the portable hole creates an extradimensional hole 10 feet deep.
The description of barrel in the rules says:
A barrel can hold 40 gallons of liquid, or 4 cubic feet of solid material.
4 cubic feet would only be about 30 gallons. I think for this exercise we need to use the liquid volume, which will more closely reflect the full space used. The 4 cubic feet for solids must assume there is a lot of empty space wasted within the barrel if you pack in solids, not that the dimensions would be smaller.
The dimensions of the barrel are also not exactly defined, but barrels are taller than they are wide. Today, a typical 40 gallon wood barrel is about 2.9 feet high, and would have a diameter of a bit more than 1.5 feet, if perfectly cylindrical. However, the belly diameter of such a barrel would be more like 2 feet, as other than modern metal barrels, wooden barrels are not really cylindrical, they are thicker around the middle. The rounded side profile of the barrel will lose you some additional space, as will the thickness of the wood. In medieval times things were not as standardized, but we'll use these dimensions as placeholders.
At these dimensions, you could pack in only three layers of barrels, and at a ratio of 3 to 1 for the thickest point of diameter, you could pack 7 barrels per layer.
This will result in a total of 21 barrels. All assuming that there is no other way to pack them denser by tumbling them in and treating them more like spheres in a cylinder.
There also this similar question on how many coins fit into a cubic foot. If you just looked at circle packing and ignored the outer dimension limits of the hole, you would get a larger result, but I think that is not realistic, the outer dimensions have a significant impact on how many barrels you can fit in.
Building on AnnaAG's answer of 21 40 gallon barrels.
I'm making the case that you could get 4 barrels in 1 layer on their side.
There is some overlap in the image but I think that can be ignored because of
If you allow that then you can stack the barrels in the following way:
