About nine years ago, before I could afford a bed for myself let alone my cat, I used to do that very thing for Oscar. Using a lid from a copy paper box just like that. I lined it with some carpet pieces that I always had access to and he loved it.
I would bet that just like what you posted here, if I were to get another box lid and put a towel in it Oscar would ditch his fancy pet bed he uses now for the box.
Patches agrees!!! he decided indoors is best this morning while a storm was raging. For a stray he sure makes himself at home. I just took this a moment ago. eventually he may become indoors only.. I could hope.