What makes a good vessel for storing sourdough starter?

I use a glass jar with a fermenting lid. In other words, vented. This jar will auto release gas or you can set the release manually.
No more worry about your glass jar breaking under pressure or anything icky falling in. I use a 1400ML or 1.5 Quart jar like the one in this link from Amazon.

I found that a wide mouth Mason jar, folded cheesecloth and a rubber band extremely successful! Success the first day!

Half gallon Mason jars with the 2 piece lid work great. Leaving the ring portion of the lid loose allows the gas to escape by lifting the flat portion. Otherwise it is sealed.

Wide mouth glass jar with a piece of wax paper loosely rubber banded over the top. Keeps it from drying out, but if the starter goes nuts, it just pushes the wax paper up and out of the rubber band--no worry of exploding glass.

Years ago when I worked with sourdough a lot I used a stoneware crock that was just the right size for the amount of starter and it had a lid that was not air tight, so it worked perfect for the job. Over the years of not having a starter (I stopped doing it when I had to go back to work) and moving a couple of times, I misplaced that "perfect" crock and it's nearly impossible to find a crock the size of the one I had, but I happened across a website that has what I think will be the right vessel for the job. It's a 1.5 Qt wide mouth glass jar, made in Italy by Bormioli Rocco and it is a really good price. If anybody is interested it's on the breadtopia website at: http://breadtopia.com/store/sourdough-starter-jar/
I have already ordered mine. Maybe this will help someone out there... I hope so.

The King Arthur crock is a nice size and looks good. The logo clearly distinguishes the contents. However, I prefer to be able to see through to the contents. A tall wide mouth glass canning jar seems ideal to me. With the ring on loose (or fitted with something breathable), I would not worry about it exploding.

If a shorter, wider container would fit your space better, Pyrex makes 7 cup glass storage containers. You can buy them with airtight, silicone-sealed glass lids, but plastic lids with vent tabs also are available.

I find wecks jars perfect. I rest the glass lids on top. So the gasses can escape if they need to. I use the straight sided ones. I like a clear jar so I can see when the bubbling starts after feeding.

If you have the kind of sourdough starter that doesn't require extra yeast, do not use glass, as it can explode. I use a crock with a spring-loaded lid. Have had my starter for over 40 years.

The starter is acid so do not use any metal. I like to use a one quart cylindrical crock which I cover with a loose fitting saucer/plate. I have been experimenting with starters of varying degrees of sourness. I keep each batch in a separate pint jar with loose fitting lid to let the CO2 out. Don't want the starter to explode out when the lid is removed. The starter is sticky so wide mouth containers are easier to extract the starter from and clean.

I use a 64 oz pickle jar and store it in the refrigerator. I take out starter out of it for recipes. Each day I put in 1/2 of my starter that I feed every 12 hrs and I use the oven light to keep the smaller jar at a constant 75 deg. I stir the larger starter jar before putting it back into the refrigerator to add air. I have had great results using this method. I tighten the lid AFTER I know there's no gases being formed.

I recommend the Harsch Fermenting Crock. I use the smallest one and it is pretty large, but I have never had a problem with my starter overflowing. The water seal on this crock lets gas out and provides a good seal. I always worry about using glass jars. You can't seal too tight or they might explode. I also don't want bread that tastes like the inside of my fridge. They are expensive and I have not been able to find an alternative. However, if you are serious about maintaining a starter I think it is worth it.

I use a regular wide mouth canning jar with an inverted coffee filter over the top, secured by a canning jar ring. It's worked great for me. if the coffee filter ends up getting stuck, as they eventually will, I just replace it with a new one. The filter breathes and also allows more yeast to get in.

Admittedly I have only stored my starter in a glass jar, but I wouldn't want anything else.

You should look for a size that is neither too large, nor too small. It should comfortably hold your sourdough starter when you are simply feeding it - I usually store 200g of starter, at 100% hydration (100g flour, 100g water) or thereabouts. However, when actually making sourdough starter you make more - perhaps 400g or so, depending on how much bread you are making, plus you want to have something left to restart your bread. A typical recipe might call for 20% starter, so 200g for a kilo loaf. The other solution is to of course have two separate jars, and only use both when doubling for bread.

When I say it must be able to hold that much, I mean that it will have to hold it fully risen, which means around 2x to 2.5x of the original size. In addition, narrow and high is better than low and wide, since it makes it easier to tell how much it rises.

Also, you will want a wide opening, so that you can easily clean it.

As for covering it, I just use a sheet of plastic. Just take care not to get it gooey. It will hold in the moisture so the sourdough doesn't dry out, but won't trap the carbon dioxide.

Good luck

as long as it's not airtight, in my experience you can use almost anything: a large jar with a loose lid, a tupperware container that has a small hole in it... that sort of thing.