Many tuners use small squeeze bottles to store finishes. One well known supplier is Harbor Freight who sell them in sets of three for less than a buck a piece. Recycled condiment (catsup, mustard, etc) bottles are cheaper and made of a thicker plastic sometimes PE, sometimes PP) and are a popular alternative. I use both.
There are two issues with plastic containers. One is whether the solvent component of the finish will attack the plastic. I've stored both shellac-based finishes (with denatured alcohol as the solvent) and lacquer based finishes (with lacquer thinner as the solvent) with absolutely no problem. I also have a polyethelene bottle of acetone in my shop, and I have an oil/wax finish (for tool handles) that is made from BLO and wax, and thinned with turpentine, so my experience is that PE is inert to both acetone and turpentine. There are other solvents that can be used instead of DNA, acetone, turpentine or lacquer thinner that I haven't tested and can't comment on.
The second issue is that plastic breathes which means that, over time, oxygen will migrate through the walls of the bottle to cause the finish to cure prematurely in the bottle. Also, the solvent can vaporize and migrate out of the bottle, resulting in a change in the dilution of the finish (this is a well-known issue with shellac-based friction polishes). Both phenomena occur faster with thinner walls, and are less of a problem with the thicker recycled condiment bottles. These are slow processes, and the tendency for it to be a problem can easily be managed by a couple of simple steps:
1. Mix finishes in fairly small quantities so that you are likely to use it up before it spoils. I use a 35mm HDPE plastic film canister as a measure when mixing finishes - it's volume is about one liquid ounce. My lacquer-based friction polish involves one equal parts of lacquer, lacquer thinner and tung oil, so the resulting mix is only about three ounces.
2. If the finish spoils after several months, just throw it (and the bottle) away - if you mix small quantities, the wastage is trivial, and the bottles are cheap.
3. To address the loss-of-solvent issue, mark the side of the bottle periodically, and if you find that after a period of not using the finish the level in the bottle has dropped below that mark, add solvent to restore the level (and dilution).
I also make a wiping varnish from equal parts of commercial alkyd varnish, tung oil, and turpentine that I use on larger turnings. This is stored in small recycled glass or plastic jars (that formerly contained cooking spices). I understand the 'instant wearout' issue with glass containers in a shop with concrete floors, but so far I have been lucky.
I also have a can of commercial solvent-based wipe-on-poly for special applications. I tend to keep it in the original pint metal can mainly because its convenient, and because the closure seems to be tighter than the cap on a Harbor Freight squeeze bottle. But I have had a can go bad after four or five years. Of course, it is easy to make your own WOP but simply diluting commercial polyurethane with a solvent (either mineral spirits or turpentine).
My experience is that water-bourne WOP lasts forever.
The related issue is that shop-made finishes tend to use commercial components, and there is a concern for how long those components survive in the original packaging. Solvents aren't a problem, but lacquers and varnishes can go bad over time. A solution that is only partially successful is to store them with the can inverted - that way, any air in the can will float above the liquid, which means that if the component cures at all, the cured portion will be toward the bottom of the can when it is later reopened.
Finally, there is the matter of glues curing in the container. My experience is that CA tends to thicken over time - thick becomes so viscous that it is unusable, while medium tends to become thick, and thin becomes medium. My solution to that problem is to purchase it in 2oz bottles, but I understand that an option if you purchase larger bottles (for economy), is to decant it to a smaller bottle for use, while keeping the larger bottle in the refrigerator.
And then there is polyurethane glue. I really dislike that stuff, but there are times when it is what the project calls for so I keep a bottle in the shop. But I have a rack that allows the bottle to be stored upside down so that if it cures in the bottle, there is usually is some liquid left about the cured level that can still be used.