It is easy to launder socks by hand. First, fill your sink with cold water and a small amount of mild laundry detergent, or in a pinch you can use dish soap, shampoo or another type of liquid soap. Give your socks a good scrunch and swish them around, then turn them inside out and do the same. Is there a cloud of muddy water coming off them? Let's hope not! Next drain your sink and rinse your socks with more cold water until they don't feel soapy any more.
To dry, you can hang your socks over a rack or on a hanger, or set them flat on something like a mesh sweater drying rack or a flat surface that gets plenty of air circulation. They should be clean and dry very soon!