We first introduce fractional Gaussian fields and their regularity properties on fractals such as the Sierpiński gasket. Building upon these, we then study the Parabolic Anderson Model (PAM) with fractional noise which is white in time and colored in space. Beyond establishing the existence and uniqueness of the solution in the Itô sense, we also obtain explicit moment estimates for the solution which lead to intermittency properties. Furthermore, our results extend to bounded domains of recurrent metric measure spaces like metric graphs and fractals.