Relic Ruins periodically use them as well.

You purchased a recipe for blacksmiths crafts. To get the TNT barrel you need 1x black powder; 3x string; 1x barrel. Or you could just hope to stumble across one in the World Market. (if the recipe is not at the blacksmiths you have to gain a few levels for it to appear)

I got mine through dock trade. It is very rare. If that fails, use gems to buy it once you have the other ingredients (at the mill where you need it)

There is a cave at the upper level of your island. Clear a path to it and blast your way in with the gunpowder barrel. It unlocks another habitant of the island to do quests for. The first batch of land costs 4000 and the second 12 500. Certain levels have had to been reached, before you can purchase them.