This is set on a fictional alien world with medieval people. However, this world has alien livestock that are capable of flying into outer space and traveling to other worlds. These medieval-age people figured out how to use the animals to tow their ships from planet to planet, all with no technology. Would they be able to create a ship or ark that can withstand the elements of outer space using common materials? They would have protection against radiation and have their own source of water and heat and air and water using fictional means. However they would need to build their ships using common resources such as oil, tar, wood, etc. Is it possible to build a ship or ark out of wood and other common materials that won't implode in space? They would also have their own type of epoxy sealant that will make their ships absolutely airtight, which would also protect them and their ships from radiation, etc.
If airtightness is the only requirement, then yes, medieval people could create such vessels
By middle ages, creating watertight ship hulls was a very common task. Yes, ships leaked, but that could be successfully mitigated.
Also, diving bells were known since antiquity, and fully isolated "dry" bell is not very different from a spaceship. 1 atm pressure difference could be easily handled by wood and metal construction.
Dealing with the coldness of space is more tricky, because iron will become brittle, and any wet insulation will become dry, but if medieval NASA can get enough tries, they can find out what works and what not.
Radiation hazard can be mitigated by building thicker hulls. This is something that the builders absolutely can do, but they must know it from somewhere, or learn the hard way.
Excessive G forces, atmospheric shocks, high temperatures and life support would probably be too much to handle for medieval technology, but that seems to be outside of scope of this question.
they can't make a airtight container big enough for weeks of travel, especially not one that can withstand vacuum. Even a day of travel is probably impossible. You need a shipping container worth of air per person per day. Since it needs to have doors it gets even less likely they can make it airtight.
The only technology they had for making airtight containers either relied on glass, or barrels, which have a scaling limit. if the travel times was an hour or few they could maybe manage with large single use barrels, assuming they someone at the other end to let them out. But even then it is iffy, wooden barrels can't withstand much internal pressure. Even the materials themselves have problems. Wood, tar, pitch, glue, all these materials outgas in a vacuum, meaning they won't work for a seal in vacuum, natural material don't make great vacuum seals. if this is actually space, then uneven heating by the sun will cause even more deformation which is destructive ot seals.
large barrels and ships are made in very similar ways at least in terms of making them watertight, tight fitting planks with either with nothing (rare) or a caulking (common) sealing joints. Usually cordage soaked in pitch or tar. All of which seriously outgas under vacuum, so they will break down or fail almost instantly when subjected to vacuum. Sealing things against vacuum is hard without modern materials because all traditional glues and sealers contain volatiles the boil off at vacuum pressures which destroys the material. If you want to get an idea of what happens put a marshmallow in a microwave. Even the water in the wood will boil off deteriorating any seals in contact with it and can even damage the wood itself.
Barrels work much better for our purposes because they are reinforced on the outside relying on compression to seal them so they can withstand some pressurization, but they still suffer the outgassing problem and the internal pressure they can withstand is still low, the end caps can't be braced like the sides so this is where leaks tend to start. Ships are pegged/bolted into place so will be far weaker to internal pressure, which makes sense they are designed to withstand great external pressure.