I'm doing a clutter item for a popular video game that's pseudo-medieval, and I have habit of trying to find solutions that are (or at least feel) era appropriate.
It's made me discover a lot of interesting things about refrigeration, the many types of clever cooking apparatuses that were available in pre-modern times, and other ingenuity that we take for granted.
I was going to make a painting can from wood or clay, and have it covered with a wooden lid or cloth and string. Googling for "era-appropriate" references proved a little difficult, so I have to assume that the way we package and store paint these days is a very modern phenomenon.
I should mention that the item I am making is supposed to evoke a can of paint for houses. As I write this, it struck me that this might - as well - be a very modern thing (to paint interior/exterior areas of the home).
Paint has two components: pigment, and vehicle. The pigment is the color, the vehicle is the liquid the pigment sits in. Many pigments in use today were in use in the middle ages, particularly the earth pigments, like the common Barn Red, an iron oxide. Typical vehicle for house paint today is two types: latex and oil. Latex paint is rubber-based, and pretty new. Oil vehicles have changed somewhat to be harder and faster drying- more like varnish. But there are various drying oils, like linseed, that have long been the basis for oil paint vehicles. Drying means oxidizing: they absorb oxygen and polymerize. When they do that, you can't re-dissolve them. Today we have paint cans that can seal the oil paint from oxygen, to ( for a while) prevent that. But that wasn't very easy in the early period- it was more practical to store a jug of linseed oil, and keg of red pigment ( or brown, whatever) and mix the paint whenever there was job. This was common even in the 19th c. Even by the later 19th c., though often the painter could have the paint merchant do the mixing, anyone who wanted to store paint for very long would tend to store the components. There was also the tendency of the pigment ( iron oxides, white lead etc. are heavy) to settle to the bottom: a container holding paint has to be stirrable. A stoneware jug might be sealable, but it was not stirrable.
Not all medieval house paint would be oil based. There was whitewash: heat something with calcium hydroxide ( like limestone, chalk, gypsum or oyster shells) to drive off the water, and it's called unslaked lime. Mix it with some water ( or, better yet, some water and a little milk) and it becomes a white paint. Once on the surface, it reacts with carbon dioxide to become harder and water resistant .This was a very common coating. It not only reflected a lot of sunlight, if you wanted the house kept cool, but the alkalinity tended to discourage insects and rot. It was not very hard and durable ( for masonry it was OK, and adding milk made it stick a little better to wood) but it could always be renewed. As unslaked lime was hygroscopic, as a dry powder it would hydrate itself from the humidity in the air after a time and become less useful, so there was a storage limit on how long you could keep it, as well. So, whitewash also would be made and mixed on demand.
A medieval house paint can would then, really, be something like a wooden bucket, maybe a piece of wooden treenware. It would hold the paint while it was being applied, but it wouldn't store it.