What's the best container to ferment vegetables?

How small? You could use mason jars...or even plastic quart containers. I frequently use Cambro brand food containers. The key is to use food safe products and avoid material that is worn or cracked, which could harbor bacteria...other than that, almost anything can work.

After searching online for weeks and coming up fruitless, I got some terrific smaller jars (2 1/2 qt) at a local Target. Glazed ceramic with wooden lids. Easy to drill a hole if you like, but I found it easier to pull off the rubber gasket which made it loose-fitting enough to breathe as-is. $15 or so.

Which container is, as you say, best for fermenting vegetables may not have a concrete, objective answer, because what is ideal for fermenting one vegetable (singular, plural, seasoned or not) may not be ideal for fermenting another. The Korean onggi however has a long history of effective use as a fermentation container owing apparently to not only its shape (which is lovely) but to its grade of porosity.

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Mason jars are ideal for small batches.

In fact, I often use multiple mason jars for larger batches rather than using my "real" fermentation rig, a large glass jar with a fermentation airlock.

Some things to know:

• Covering the jar. Some people lay a cloth over the top. I much prefer plastic mason jar lids. During fermentation you can leave them screwed on loosely to let CO2 vent, and since they're solid they keep out dust, insects, mold spores, etc. Once you're done fermenting, just screw them on tight to refrigerate. (Metal lids sometimes corrode from splashed brine.)
• Light. You'll read in some places that glass is bad because it lets in light—that you need to keep mason jars in a cupboard or closet. I keep mine on the kitchen counter. In a shaded spot, not direct sunlight, but still in the open. Fermentation still happens. Bonuses: they're pretty to look at, a good conversation piece, and seeing them reminds you to check them.
• Submersion. Big crock systems have the great benefit of a sinker that keeps the vegetables submerged. It's trickier with a small-necked mason jar. I've heard (and tried) many techniques. Simplest by far: cut a carrot into thin "planks", like little wooden boards, just a little wider than the jar's mouth. Nestle them on top of the vegetables in a cross pattern. Except for very small things (like bits of cabbage and the occasional peppercorn), they hold everything down. It's easy to skim off any yeast or mold, and you can eat them afterward.

For the covering and submerging parts, many people use a small plastic zip lock bag filled with salt water for small and large mason jars. I'm not a fan:

• Fussy. When scum forms, it's all around the bag—so you're not just skimming, you're also removing, washing and replacing the bag.
• Safety? Though plastic bags are food-safe, I'm uncomfortable leaving them floating in an acidic, saline brine for days or weeks at a time. They probably don't leach chemicals into the brine... but carrots definitely don't.