We show that contrary to anticipation suggested by the dictionary between rational maps and Kleinian groups and by the hairiness phenomenon", there exist many Feigenbaum Julia sets $J(f)$ whose Hausdorff dimension is strictly smaller than two. We also prove that for any Feigenbaum Julia set, the Poincaré critical exponent $\delta_\mathrm{cr}$ is equal to the hyperbolic dimension $\operatorname{HD}_\mathrm{hyp}(J(f))$. Moreover, if $\operatorname{area} J(f)=0$ then $\operatorname{HD}_\mathrm{hyp} (J(f))=\operatorname{HD}(J(f))$. In the stationary case, the last statement can be reversed: if $\operatorname{area} J(f)> 0$ then $\operatorname{HD}_\mathrm{hyp} (J(f))< 2$. We also give a new construction of conformal measures on $J(f)$ that implies that they exist for any $\delta\in [\delta_\mathrm{cr}, \infty)$, and analyze their scaling and dissipativity/conservativity properties.