I have a pair of socks that leave marks on the skin. Since I can't throw them away, I need a solution that helps me wear these socks without leaving any mark and feel comfortable as well. Any suggestions?
Although, for a more left-field answer...
If the elastic is still too tight for the rolling method described by James, you could always break the elastic and find an alternative way to hold your socks up.
Sock Glue exists, and is intended mainly to hold up long (overknee) socks. It's surprisingly widely used - from professional dancers to Japanese girls looking to maintain the "absolute territory" effect.
It attests to being long-lasting, and apparently leaves no marks when washed off.