Couch has quite thick roots with sharp spikes at the ends. These can push straight through a growing potato so are not held back by weed suppressing fabric or cardboard - sorry! The roots also go quite deep and the only way, other than pouring poison all over your food growing area, is to dig the whole area, and the surrounding area too, deeply, removing every section of root.
However, if you have lots of fine hairlike roots that doesn't sound like couch, maybe more like nettles, especially if they are yellowish. The same applies though - they have to be dug out, but nettles don't grow so deeply as couch. They do set seed though. Fine white roots throughout the soil could also be creeping thistle - you have a battle on your hands.
First thing to do is to make a positive identification of the weeds involved. This is fairly difficult to do at this time of year, until they start to push up some top growth.
Then you might need to dig a whole lot deeper and wider (all three of these weeds will spread sideways), then start again with fresh soil, bought in (but carefully chosen) or sieved molehill soil with good, nettle-seed free compost. It will be hard work, but well worth it in the long run.