Using first-principles calculations and atomistic spin simulations, we predict stable isolated skyrmions with a diameter below 10 nm in a monolayer of the two-dimensional van der Waals ferromagnet Fe$_5$GeTe$_2$, a material of significant experimental interest. A very large Dzyaloshinskii-Moriya interaction (DMI) is observed due to the intrinsic broken inversion symmetry and strong spin-orbit coupling for monolayer Fe$_5$GeTe$_2$. We show that the nearest-neighbor approximation, often used in literature, fails to describe the DMI. The strong DMI together with moderate in-plane magnetocrystalline anisotropy energy allows to stabilize nanoscale skyrmions in out-of-plane magnetic fields above $\approx 2$~T. The energy barriers of skyrmions in monolayer Fe$_5$GeTe$_2$ are comparable to those of state-of-the-art transition-metal ultra-thin films. We further predict that these nanoscale skyrmions can be stable for hours at temperatures up to 20 K.