Consider the linear stochastic fractional heat equation with vanishing initial condition: $$ \frac{\partial u (t,x)}{\partial t}=-(-Δ)^{\fracα2}u (t,x) + \dot{W}(t,x),\quad t> 0,\, x\in \mathbb R, $$ where $-(-Δ)^{\fracα{2}}$ denotes the fractional Laplacian with power $α\in (1,2]$, and the driving noise $\dot W$ is a centered Gaussian field which is white in time and has the covariance of a fractional Brownian motion with Hurst parameter $H\in\left(\frac {2-α}2,1\right)$. We establish Chung's law of the iterated logarithm for the solution at $t=0$.