Short answers, pulled from the story.
Volume is a measure of regions in three-dimensional space. It is quantified using SI derived units such as the cubic metre and litre, or imperial and US customary units such as the gallon, quart, and cubic inch.
The earliest evidence of volume calculation comes from ancient Egypt and Mesopotamia, written as mathematical problems approximating shapes such as cuboids, cylinders, frustums, and cones. These appear in the Moscow Mathematical Papyrus, dated to around 1820 BCE.
Archimedes probably did not submerge the golden crown because of the extreme precision required. He is thought to have used a primitive hydrostatic balance, weighing the crown against a chunk of pure gold of similar weight while both were submerged underwater.
The metric system was formally defined in French law on the 7th of April 1795 using six units, three of which relate to volume: the stère of one cubic metre for firewood, the litre of one cubic decimetre for liquids, and the gramme defined by the mass of one cubic centimetre of water at the temperature of melting ice.
Capacity is the maximum amount of material a container can hold, measured in volume or weight, while volume is the space the material occupies. A container is bound by physical volume rather than weight, so a 50,000 barrel tank that holds 7,200 metric tonnes of fuel oil cannot hold the same mass of less dense naphtha.
The United Kingdom's Weights and Measures Act 1985 makes one imperial gallon precisely equal to 4.54609 litres, with no use of water. The earlier 1824 definition had set the imperial gallon as the volume occupied by ten pounds of water at 62 F.