In order to answer part of what you wrote in the first paragraph, the pool is 90 inches in diameter at the outer edge. It does not give the interior diameter, but we can get a rough estimate by looking at the cylinders that make up the wall. Three cylinders add up to 20 inches, so each cylinder is roughly 7 inches in diameter, which makes the interior diameter of the pool 76 inches (radius = 38 inches).

Calculate the volume:
38² * pi * 20 = 90,729 in³ = 392 gallons. 38² * pi * 14.5 = 65,779 in³ = 284 gallons

I get a different number when calculating for the ball size. You use a radius of 1.08 inches. I calculate 5 cm /2 = 2.5 cm radius = 0.98 inch diameter. I also would like to point out that a 5 cm ball is smaller than the balls in a commercial ball pit (think Chuck E Cheeses). Standard ball pit balls would probably be more like 3 inches in diameter, but that's a rough guess from memory. I don't have anything to measure.

Using the packing density of 0.64 you give from XKCD, we get the following:

5cm balls: 0.98² * Pi * 4/3 = 4.0 cubic inches 65,779 in³ * 0.64 / 4.0 in³ = 10,525 balls

3in balls: 1.5² * Pi * 4/3 = 9.4 cubic inches 65,779 in³ * 0.64 / 9.4 in³ = 4,479 balls

Here are a couple of sources for 3 inch balls:

26 cents per ball http://www.amazon.com/150-Large-Crush-Proof-Ball-Balls/dp/B000VX2PO0/ref=sr_1_3?s=toys-and-games&ie=UTF8&qid=1449891804&sr=1-3&keywords=3+inch+ball+pit+balls

21.31 cents per ball http://www.amazon.com/Pack-Crush-Proof-Ball-Balls/dp/B000NNM41C/ref=sr_1_6?s=toys-and-games&ie=UTF8&qid=1449891804&sr=1-6&keywords=3+inch+ball+pit+balls

That's still super expensive (4,479 * 0.2131 = $954.47) and costs more than the 5 cm balls (10,525 * 0.082 = $863.00), though the 5 cm balls may not give the same ball pit experience you remember as a kid. Getting a smaller pool would definitely make it more affordable.

Edit: You could probably get by with significantly fewer balls depending on the age of your kids. Here is a small ball pit without balls, but take a look at the dimensions and number of balls in the first review.

http://www.amazon.com/FocuSun-Playpen-39-4-inch-19-7-Inch-Zippered/dp/B00XPLB35M/ref=sr_1_1?ie=UTF8&qid=1449892883&sr=8-1&keywords=4+inch+ball+pit+balls