Caretaker of cancer patient

I doubt such a thing is covered by any health insurance. I think there are some special programs set up by independent groups that do this for the parents of minor children getting certain treatments; one sponsor is McDonald's with their Ronald McDonald houses in various German cities. The closest I see on their website is in Mainz which is around 40 Km away from Frankfurt but well connected by trains. But again this is just for parents of children and might even be limited to having treatment at particular facilities. I am posting a link to the site about this program although I doubt it will fit your needs. What is provided by health insurance is taxi transportation for a patient from home to a hospital and back - and I assume there is no extra charge for someone to accompany them.

https://www.mcdonalds-kinderhilfe.org/

The McDonalds houses (Thanks, Tom, for posting about them!) are a great initiative by the fast-food chain and helped our good friends greatly when their newborn, due to a serious liver problem, had to stay in a faraway specialist hospital for six months (!). The father gave up his job (and income) to be with baby and mother (only she could stay in the hospital with the kid). If not for the McDonalds house, this would have meant bankruptcy for them. In the end, he donated part of his liver to save the girl.
But yes, this is only an option for parents of hospitalized kids.
You could try a hostel, Couchsurfing or similar organisations.