  We define linear stochastic heat equations (SHE) on p.c.f.s.s. sets equipped
with regular harmonic structures. We show that if the spectral dimension of the
set is less than two, then function-valued "random-field" solutions to these
SPDEs exist and are jointly H\"older continuous in space and time. We calculate
the respective H\"older exponents, which extend the well-known results on the
H\"older exponents of the solution to SHE on the unit interval. This shows that
the "curse of dimensionality" of the SHE on $\mathbb{R}^n$ depends not on the
geometric dimension of the ambient space but on the analytic properties of the
operator through the spectral dimension. To prove these results we establish
generic continuity theorems for stochastic processes indexed by these
p.c.f.s.s. sets that are analogous to Kolmogorov's continuity theorem. We also
investigate the long-time behaviour of the solutions to the fractal SHEs.
