We study the linear stochastic fractional heat equation $$ \frac{\partial}{\partial t}u(t,x)=-(-Δ)^{\fracα2}u (t,x)+\dot{W}(t,x),\ \ t> 0,\ \ x\in\RR, $$ 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 $H\in\left(\frac {2-α}2,\frac 12\right)$. We establish exact asymptotics for the solution as both time and space variables tend to infinity and derive sharp growth rates for the Hölder coefficients. The proofs are based on Talagrand's majorizing measure theorem and Sudakov's minoration theorem.