Short answers, pulled from the story.
The Moscow Mathematical Papyrus was written around 1820 BCE and contained mathematical problems approximating the volume of simple shapes such as cuboids, cylinders, frustums, and cones. Ancient Egyptians devised units of volume based on their units of length including the cubit, palm, and digit to create the volume cubit, volume palm, and volume digit.
Archimedes principle is a method to calculate the volume of an irregular object by submerging it underwater and measuring the difference between the initial and final water volume. Archimedes likely devised a primitive form of a hydrostatic balance where the crown and a chunk of pure gold with a similar weight were put on both ends of a weighing scale submerged underwater.
The metric system was formally defined in French law on the 7th of April 1795 using six units including the stère for volume of firewood, the litre for volumes of liquid, and the gramme for mass. The gramme was defined as the mass of one cubic centimetre of water at the temperature of melting ice.
Bonaventura Cavalieri applied the philosophy of modern integral calculus to calculate the volume of any object around the early 17th century. Cavalieri's principle states that using thinner and thinner slices of the shape would make the resulting volume more and more accurate.
Commonly used metric units for volume include the cubic millimetre, cubic centimetre, cubic decimetre, cubic metre and the cubic kilometre. The conversion between these units follows a factor of 1000 where 1000 cubic millimetres equals 1 cubic centimetre and 1000 cubic centimetres equals 1 cubic decimetre.