Nothing worthwhile in Grounded is easy to get, and this applies to berry leather. You can only get berry leather from, unsurprisingly, berries, and they can only be found in one location.
Where to find berries
You can find berries on the southeast side of the map, show above. Just like the Oak Tree, the Berry Tree is a hive of bugs, spiders, and threats. It is advisable to not go to this area unless you really need to. Spiders live all around the base of the tree, so make sure you have a bow, plenty of arrows, and some armor.
It is also a good idea to set up a Lean-To nearby, set it as your spawn point, then venture through the last part of the journey to try and get the berries. You will be able to see the berries up in the tree, and can shoot them down using arrows, then use your axe to chop them up.
Loot the Berry Chunk that drops, then you need to make your way to an Analyzer. The good news is that there is another Field Station close by, shown on the map above by the purple icon.
Here, you can analyze the Berry Chunk, and then you will get the recipe for Berry Leather, but you will need a Workbench to make it. Make sure you grab as many berries as you can when you are at the Berry Tree, as you want to limit how many times you need to go into that area. It is one of the business in the game when it comes to bugs that want to kill you, and is best avoided when possible.