I'm still a work in progress, i have some stubborn areas that I'm still working with. That being said, I'm about 100% better than were I was just 4 months ago. So, I can't really claim to know "the answer", but, I have been dealing with this all my life, so I know a thing or two about it.
I was born with a rash on my hand, later it moved to my face, behind my ears, my nose, bends of knees and elbows... developed allergies later, lots of fungal infections all my life, then chronic sinusitis. I used typical mainstream treatment most of my life (30 years), until I got into holistic approaches about 4 years ago out of desperation. I did lots cleansing (28 liver flushes, countless colonics and colon cleanses, kidney cleanses), cleared candida overgrowth and generally improved my health, but, the eczema actually got worse (its called unsuppression in homeopathy). finally, with eczema on my head, face and hands that looked like third degree burns, I tried homeopathy. I had always ignored homeopathy because, like you, I was skeptical on scientific grounds, and assumed it was placebo at best. when I tried it, I didn't expect it to work. but, it began to work almost immediately, and not just on my skin, I improved in mood, attitude and general health.
in homeopathy there are probably over 30 remedies that could apply to your eczema, and for it to work, the correct remedy must be chosen. for this reason, it takes some careful thought or professional help before starting. to get a really good understanding of how it works, there is a book, "the homeopathic treatment of eczema" by Logan, which you can get on amazon. check it out, its very helpful and you might even be able to pick your remedy based on the info in the book.
also, there is a forum called "abc homeopathy", you can google it... post your detailed history, and see if you can get some help. the only caveat is, you have to research the advice given, because not everyone who offers advice on the forum knows what they are talking about.