Does eczema ever go away?

Yes and no. Eczema, the symptom, can go away with treatment, but eczema, the disease, does not go away but is only controlled.
Sometimes. Eczema is a genetically-based skin condition, which has no known cure. There are treatments, however, including prescription and over-the-counter creams, as well as allergy-based approaches. Most people enjoy at least some improvement from treatment, and in some people the eczema goes away, at least for some time.