We study the limit distribution of the volume fraction estimator $\widehat p_{λ, A}$ (= the Lebesgue measure of the intersection $\mathcal{X}\cap (λA)$ of a random set $\mathcal{X}$ with a large observation set $λA$, divided by the Lebesgue measure of $λA$), as $λ\to \infty$, for a Boolean set $\mathcal{X}$ formed by uniformly scattered random grains $Ξ\subset \mathbb{R}^ν$. We obtain general conditions on generic grain set $Ξ$ under which $\widehat p_{λ, A}$ has an $α$-stable limit distribution with index $1 < α\le 2$. A large class of Boolean models with randomly homothetic grains satisfying these conditions is introduced. We also discuss the limit distribution of the sample volume fraction of a Boolean set observed on a large subset of a $ν_0$-dimensional $(1 \le ν_0 \le ν-1$) hyperplane of $\mathbb{R}^ν$.