How committed are you to the wooden bins? If they are on display and you like them for aesthetic reasons, I can relate. My house is full of choices like that.
But of the possible options, wood is the most difficult in practical terms. Wood dries out and warps, or absorbs moisture when washed (and warps), so it needs to be treated with something that is food-grade and won’t leach into and flavour the contents. For food-contact items like cutting boards or salad bowls I like a mineral oil or beeswax finish, but these are items I can quickly wipe off. For long-term storage of things like flour or sugar the inside of an oiled wooden bin would become coated with a sticky mess that would be hard to clean off. If the bin is constructed like a box, the inside corners are probably small-radius and hard to get at. I agree with @ChickieD’s suggestion of a liner, with the addendum that I like these clips much better than twist ties.
Pottery jars have been around forever, and still work. See if you can find Grandma’s kitchen canister set (hint: the one at the garage sale may not be your grandma’s, but it’s somebody’s grandma’s).
Since tinware became available centuries ago, it pretty well took over from organic containers. There are still lots of cake and cookie tins with press-on lids at the dollar store, or see Grandma above.
Example, but maybe not exactly what you’re looking for:
My personal choice for most storage is glass, from spice jars up to suitable sizes for sugar, coffee, etc.
We use a lot of these
or mason jars. You can get plastic lids for mason jars that are easier to use than the canning lids.
Glass is easy to clean, seals really well, and you can see the contents, but I keep the jars in a drawer or cupboard away from light.
We buy flour in fairly large quantities, so I will confess to using plastic for the bulky stuff. Plastic is unbreakable, lightweight, which becomes an issue if you are talking about 3-liter containers or larger, and it doesn’t make a loud crash when you put it down on a hard counter. It’s terminally unaesthetic and uncool, but I have given up that struggle.
The best of both worlds might be if you can find tin or plastic containers that fit reasonably well inside the wooden bins.
(I apologize for not linking to Monty Python’s classic piece on storage jars, but some spoilsports seem to have blocked it on YouTube.)