Where should I take my old white goods if they’re too far gone to fix?

What are white goods? 

Any thing made from fabric - clothing, linens - should toss in the trash if it is that worn.

Any large appliance must be carted off to the dump or scrap metal recycling or pay to be picked up by your trash company.  There are junk pick up companies that specialize in washers, dyers, fridges. 

Wrong category. 

The council should take them away, there is a charge though, or you could contact a scrap metal dealer. 

I'm not familiar with the term "white goods." What does it mean?

If it's the same as bed linens, if they're too far gone for repair, they cannot be donated. If they're 100% cotton, they can be torn into small pieces and composted, but other than that, they'll land in the trash, then a landfill.

Maybe someone with sewing or crafting skills could use them? Lots of home sewing people make a practice garment, called a muslin, to deal with fitting issues before they use their real fabric. Old sheets would be fine for that purpose.

It depends on what country. lf you live in the UK, you can contact the refuge department of your local council, they sometimes collect old televisions, washing machines, fridges etc.