Reverse engineer it from known metal densities (and marvel at how tiny and dense the currency is)
Given we know the standard weight of all currency as described in 5e D&D (0.32 oz or 9.07 grams, as there are 50 coins to a pound regardless of denomination), we can reference the metal density to find out what the volume of coinage is, and therefore work out how much space a given value of coins must occupy.
- Copper - 0.324 lb/in3 - ~0.062 in3/coin - 27,993.6 coins per cubic foot
- Silver - 0.379 lb/in3 - ~0.053 in3/coin - 32,745.6 coins per cubic foot
- Gold - 0.698 lb/in3 - ~0.028 in3/coin - 60,307.2 coins per cubic foot
- Platinum - 0.775 lb/in3 - ~0.026 in3/coin - 66,960 coins per cubic foot
Since pure trade bars and the actual currency have the same value per weight, I am assuming that the coins are pure and have the same densities (but, as explored here, this is not necessarily consistent with how currency has been described or depicted in D&D's history). The values given here also don't account for the fact that the coins are probably not shaped so that they perfectly tessellate. As this blog post from dmsworkshop.com helpfully summarises, assuming round coins that are about 1/16th of an inch thick and roughly 1 inch in diameter (varies depending on denomination):
To save you some math, the ideal packing density of coins is 78.6% (if neatly ordered in stacks) or about 60% if loose. This means that an unorganized heap of coins (like those stuffed into a sack) will contain about 40% empty space.
Non-round shapes would improve the packing density to varying degrees depending on the exact shape of the coin, but using round coins seems a sensible default. So taking that into consideration:
- Copper - ~22,000 coins per cubic foot (neatly stacked) - ~16,800 coins per cubic foot (loose jumble)
- Silver - ~25,740 coins per cubic foot (neatly stacked) - ~19,650 coins per cubic foot (loose jumble)
- Gold - ~47,400 coins per cubic foot (neatly stacked) - ~36,180 coins per cubic foot (loose jumble)
- Platinum - ~52,630 coins per cubic foot (neatly stacked) - ~40,180 coins per cubic foot (loose jumble)
Using these values we can calculate that, for instance, a coffer filled to the brim with 1,000 neatly stacked gold coins would be slightly over 36 cubic inches in capacity (for instance, 6 inches × 6 in. × 1 in. internal dimensions), or loosely piled somewhat less densely at about 48 cubic inches (for instance, 6 in. × 6 in. × 1.33 in.).
You're probably noticing that those are very small dimensions. Metals, especially precious metals, are dense, and sensibly shaped coins made of these metals are small - much smaller than we are apt to imagine them (see Brythan's answer for a helpful comparison to US coinage). Even if you use a chest loosely filled with 10,000sp (equivalent value to 1,000gp) that only comes to about 880 cubic inches - roughly 1 ft. × 1 ft. × 6 in. internally, but weighing in at a hefty 200 lbs!