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.