  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.
