  We present new experimental low-temperature heat capacity and detailed
dynamical spin-structure factor data for the quantum spin liquid candidate
material Ca$_{10}$Cr$_7$O$_{28}$. The measured heat capacity shows an almost
perfect linear temperature dependence in the range $0.1$ K $\lesssim
T\lesssim0.5$ K, reminiscent of fermionic spinon degrees of freedom. The spin
structure factor exhibits two energy regimes of strong signal which display
rather different but solely diffuse scattering features. We theoretically
describe these findings by an effective spinon hopping model which crucially
relies on the existence of strong ferromagnetically coupled triangles in the
system. Our spinon theory is shown to naturally reproduce the overall weight
distribution of the measured spin structure factor. Particularly, we argue that
various different observed characteristic properties of the spin structure
factor and the heat capacity consistently indicate the existence of a spinon
Fermi surface. A closer analysis of the heat capacity at the lowest accessible
temperatures hints towards the presence of weak $f$-wave spinon pairing terms
inducing a small partial gap along the Fermi surface (except for discrete nodal
Dirac points) and suggesting an overall $\mathbb{Z}_2$ quantum spin liquid
scenario for Ca$_{10}$Cr$_7$O$_{28}$.
