The other thing I use for chains besides old socks is sections of blue jean legs. These are pretty durable and I write the chain size on them with magic markers. I carry my saw tools, files, spare chains, etc in an old metal Sawzall case. I rarely am working more than a few feet from my truck so don't need to pack anything through the woods. I am mostly working in people's yards and not as much in the forest. If I have to carry wood over about 30 feet, it usually stays where it is.
In the shop where I have a large supply of chains of all lengths and sizes, I keep them in clear plastic shoe boxes by type (gauge and pitch) and put ones of specific lengths in jean legs to keep them separated. ie, I have a box of 3/8 x .050, 3/8 x .058, .325 x .050, .325 x .063, 3/8 x .043, etc etc. Many of these chains don't fit bars I have. some day I'll find a use for them. I keep these shoe boxes in file cabinets. I have dozens of file cabinets in my shop for tool and hardware storage. They're cheap and work great.
Bars hang on S hooks on the wall and are labeled with duct tape with mount type, pitch, gauge and dl count. I have a number of those I don't have saws for too. Someday...