The problem of Lebesgue area of polynomial Julia sets goes back to classical work of Fatou. When the polynomial is reasonably hyperbolic then the Julia set has zero area, and there are plenty of examples of this kind. First examples of polynomial Julia sets of posiive area were constructed around 2006 by Buff and Cheritat. However, these examples are rare and topologically ``wild". In the talk, we will describe a new class of examples, certain ``Feigenbaum Julia sets", that are ``tame" and ``observable", with explicit topological models and computable images. An extra curious feature of these examples is that their hyperbolic dimension is strictly less than the Hausdorff dimension. The corresponding parameter set has a reasonable size (positive Hausdorff dimension). On the other hand, we can construct examples of Feigenbaum Julia sets whose Hausdorff dimension is strictly less than two. Existence of such Julia sets goes against intuition coming from hyperbolic geometry and theory of Kleinian groups. It is a joint work with Artur Avila.