Is it possible to have aromatic rings inside a structure that itself is non-aromatic? Is there an example.
Sounds like a contradiction in terms to me. If a compound has aromatic ring(s), then it is aromatic - that's the definition. Though aromatic compounds with aliphatic side chains are possible, of course.
Your compound is azulene; look it up on Wikipedia or elsewhere. Indeed, it is aromatic, and so is each of its rings. You may be confused by the fact that cyclopentadiene (which looks just like one of these rings) as an individual compound is not aromatic, but that's another story. See, a 5-membered ring needs an extra electron to be truly aromatic. At the same time, the 7-membered ring has one electron more than it needs. So they exchange that electron and live happily ever after.