I am pretty reliant on the washing machine and dryer right now. If there were no electricity I would not have a problem hanging a clothesline. I would use some paracord and just string it across two solid objects.
Washing by hand would be a lot harder. You'd need a bucket, detergent and water. But your hands sure will get tired and pruney.
I ran across this post about the Hillbilly Washing Machine and I thought it was neat. I already have a bucket but I'd have to get a new plunger from Walmart.
If you have a secure lid for the bucket another thing you can do is just soak the dirty clothes in the bucket (like colors only), then snap on the lid. First check if it leaks. If the lid is snug, roll the bucket around and most of the clothes should come out clean after a few minutes - low tech agitating action! Wring all the water out and hang to dry.