In the post-apocalyptic realm, trees are near extinct and metal/plastic is hard to come by and better used to repair things or make weapons. Other materials, like leather, bone, or wool, are still relatively common
What sort of materials could be used to make barrels in this scenario? The barrels are made to hold both liquid and solid products. The barrels should be traditional barrels with a bulge if such could be done without wood
You can try to reinforce these with wood or reeds, but rely on the clay to actually form the watertight seals. The downsides include increased risk of breakage and greater weight.
It is possible to sew leather together to form watertight containers. Bison, cow, and other large animals can get you up to scale pretty quickly. It would be difficult to transport and carry, but even a wicker basket could help contain it and provide much needed hand-holds.
I should note I say "bladders," but this is more of a quirk of English not distinguishing between bags for holding liquids and the organ. You could take leather skins and make a "bladder" out of it, no urinary system involved!
Tanning is also important in this application. It gives a bit of durability to the structure and helps with waterproofing. Best of all, you only need urine to transform a skin to a bit of leather. It's been a historically stinky but useful process.
Worked fine for thousands of years.
Many clays will give acceptable performance even if unfired. Most will be better fired, of course. Terra cotta needs only a little heat to go hard and (mostly) waterproof.
For liquids, you could branch out and try amphorae.
Exotic options include glass or porcelain/high fired ceramics.
Beyond that, it's leather, straw and tar (Moses style), or carving out rock (using the metal you lack).
Amphorae
Ancient people loved big jugs. They are great for holding wine and grain and water. They made the big jugs from clay gathered from the riverbed and baked in a kiln.
Plus they are better for displaying saucy art than a boring old wooden barrel with metal hoops. They even have curved sides like you want!
Some of these jugs were big enough to live in. See my main man Diogynes:
Get out of my light, Alexander you twat!
Whoops that's a wooden tub.
Perfect.
Woven baskets. With a tight double-weave, you can make a basket that will hold water. Several Native American tribes made these for carrying water. A double-woven basket looks almost like one basket inside another basket. To get them water-tight requires a very tight weave. Some will be made with reeds from plants frequently (and conveniently!) located adjacent to wetlands.
Wicker is the oldest furniture making method known to history, dating as far back as 5,000 years ago. It was first documented in ancient Egypt using pliable plant material, but in modern times it is made from any pliable, easily woven material.
The sort of wicker that you are most likely familiar with is an open weave.
For long-term storage of water (or other liquids), especially where your culture is stationary, I would strongly recommend pottery. While wicker is an ancient technology, archeological evidence of pottery extends to at least 25,000 BC.
Samples of water-tight woven baskets:
The first 2 are from what is now Washington state:
For carrying water. Note that there are 4 handles - a large "bag" of water is very heavy! One cubic foot of water weighs 62 pounds.
For boiling water (fill about 3/4 with water, then place heated rocks inside).
From the Four Corners region, smaller water "bottles".
A Cherokee water-tight basket from what is now Oklahoma.
This person shows some of the skills you would need to learn to make your own double-woven basket. There are 4 YouTube videos: 1 (making a tool you need), 2 (making cordage to wrap your wicker - or make replacement shoelaces - but that's a different video in this series), 3 (part 1 of making the basket), and 4 (part 2 of making the basket). In these videos, he uses yucca. In others he uses cattails (also called "reed" or "bulrush").
Just to add onto the other very good answers already here, water skins are already often made from leather, so scaling them up and making a barrel supported by a bone chassis with leather on the inside and outside sounds like a good-enough analogue for a barrel. The bones would likely have to be joined and kept together with something like clay or some other adhesive to maintain integrity. For things that aren't liquid, baskets weaved from reeds and other materials are also a perfectly good container as well.
Gourds, which are related to pumpkins, have long been used as containers. They grow in many shapes and sizes.
These plants were domesticated in the America's and have since then spread around the world as a plant that can be dried to create a containers, for example this wine bottle above from China. A gourd can either be dried over the winter or purposely dried in an oven or in the sun. Then the inside of the plant is cleaned and water proofed for example with bee's wax.
You won't get the bulge, but geengineered bamboo could well be part of the cause of your apocalypse. It's already an invasive species and a genetically modified/engineered variant might spread uncontrollably wiping out the majority of other flora on the planet.
Concrete.
Concrete has been around since at least Roman times and probably before. Roman era concrete was actually very good by even modern standards. https://en.wikipedia.org/wiki/Roman_concrete
you could I think conceivably create a clay form to pour into, or even use clay bricks mortared together with concrete and lined with concrete.
Concrete barrels are currently used to age wine as one example.
It would be a complex solution and probably over-engineered but as far as the scenario goes, I think it would be conceivable given concrete's ancient origins.
Entire preserved animals.
https://www.pinterest.com/pin/783063453946791043/
This zebra is actually full of beer. Its body cavity provides the barrel-like staves, interior to its waterproof hide. Tap is not shown. Other preserved animals are made into barrels of various sizes to contain dry goods, dwarves and hobbits, fish for shooting, fun monkeys and other objects typically contained in barrels.
Ice can be used if the surrounding temperature is appropriate and the material stored is still liquid at that temperature.
Some liquids like salt water would destroy the ice bottle over time, but maybe not all and it may still work for the short enough time with thicker walls.
Plastic was hard to come by until your survivors realized that there are many cubic miles of garbage dumps waiting to be harvested. Because "Memories are fragile — garbage lasts forever". Many plastics are thermoplasts and are easily formable at temperatures much easier achieved than the ones for smelting ores or forming metals.
Settlements started to grow next to the largest dumps, and trade in valuables scavenged from the dumps spanned entire continents, as it did with spices, gold, amber and jewelry.
why not vessels made from stone?
we have stone vessels that would last many years. the problem with these stone is it is difficult to move and build but would be still useable.
