Because I don't have a great deal of silk floss I store mine in a special box. The leftover thread I store in small bags, with the thread tag, so I don't get muddled up.. Be interesting to see how other stitchers store their silk floss....
If it came as a skein, like six strand cotton, I'd use a stitch bow or a baggie. I'd be reluctant to wind it on a bobbin because it's difficult to get the kinks out later.