I was able to do something similar by using gross approximations and conversions to/from metric. My coworker had just bought a surplus stainless steel water tank for solar heating, and was wondering how much it would weigh when full. It was cylindrical, so I asked him for the diameter and the height. In my head I converted those measurements to inches, then to centimeters by multiplying by 2.5. I divided the diameter by 2, squared it, and multiplied by 3 (close enough to pi) to get the area. Then I converted the area and the height to their nearest power of 2 so I could take advantage of logarithms. Multiplying the area and height was as easy as adding the exponents, which gave me cubic centimeters. The weight of water is almost by definition 1 gram per 1 cc. Divide by 1000 to get kg by subtracting 10 from the exponent, then multiply by 2 to get approximate pounds by adding 1 to the exponent. By the time he was done telling me the dimensions, I had an answer for him. It definitely wasn't correct, but all he needed was a ballpark anyway.

> he was able to work it out in his head, and pretty quickly

That's a scary ability. I wonder if that guy is equally adept at converting acre-feet to gallons, or if he's a one-trick pony.

Here's what I know without looking anything up: 1 mile = 1760 yards, 1 yard = 3 feet, 1 foot = 12 inches; 1 US gallon = 231 cubic inches (exact conversion, and a weird number at that). So 1 cubic mile = (63360 inch)^3 = 7707820032000/7 gal ≈ 1.101 trillion US gallons.

Fun fact, combustion engine displacements used to be quoted in cubic inches. Note that 61 in^3 ≈ 1 litre.

It's not too difficult to know or calculate inches in a mile:

    12 * 3 * 1760 = 12 * 5280 = 63,360

A few Americans may know how many cubic inches in a US gallon:

    231 

But I don't believe anyone (apart from rainman/prodigy circus performers) can do this in their head:

    63360^3 / 231 = 1,101,117,147,428.571

Estimate yes; calculate no.