Let $X$ be any irreducible scheme with the property that the generic point $\eta$ of $X$ is an set open wrt underlying Zariski topology.
What can we say about the structure of such schemes? Topologically, algebraically,…
(note, that I'm not assuming $X$ to be Jacobson, what would imply that every open subset contains a closed point of $X$ ( compare to the proof in 33.20.3 (3) in https://stacks.math.columbia.edu/tag/0A21), since in this case this would trivially imply that $X$ is zero dimensional)
Edit#1: Let follow Martin Brandenburg's suggestion to assume for sake of simplicity $X=\operatorname{Spec}R$ to be affine.
#Edit#2: If we moreover pass to reduced ring, the question becomes about the structure of rings which become fields after beeing localized at a single element.
Best Answer
Let's assume $X = \mathrm{Spec}(R)$ is affine. Since $X$ can be replaced by $X_{\mathrm{red}}$, let's also assume that $X$ is reduced. Then $R$ is an integral domain. The condition that $\{ \eta \}$ is open means that there is some $f \in R$ with $\{ \eta\} = D_f$. So $D_f$ is an integral affine scheme with exactly one point, so it must correspond to a field. But then $Q(R) = R[f^{-1}]$. Conversely, if there is some $f \neq 0$ such that $Q(R) = R[f^{-1}]$, then $D(f) = \{ \eta\}$.
The condition $Q(R) = R[f^{-1}]$ is equivalent to: every non-zero element of $R$ divides some power of $f$. If $R$ is factorial, this means that $R$ has at most one prime element (up to a unit). I am not sure what else to say. Of course, the most "famous" example is that of a DVR.
The same question has been asked at MSE: