We define and study fractional stable random fields on the Sierpiński gasket. Such fields are formally defined as $(-Δ)^{-s} W_{K,α}$, where $Δ$ is the Laplace operator on the gasket and $W_{K,α}$ is a stable random measure. Both Neumann and Dirichlet boundary conditions for $Δ$ are considered. Sample paths regularity and scaling properties are obtained. The techniques we develop are general and extend to the more general setting of the Barlow fractional spaces.