In his paper Toward a Numerical Theory of Ampleness, Kleiman writes in the introduction
[...] M. Nagata [17] and H. Hironaka [9] and [10] gave pleasant examples of complete non-singular non-projective threefolds. Each of these varieties has two or more points which are not contained in any open affine subset; whence, it is not projective.
I don't understand how a point can not be contained in any open affine subset. Is that not the definition of an abstract variety, that every point has an affine neighbourhood?
Or does he mean that there are two points, and they never belong to the same affine open? This would make sense, because in projective space every two points are contained in an affine open, because we need to choose a hyperplane, that does not meet both points.
As noted in the comments, it should mean the non-projective threefolds contain points two points $x, y$ which are not contained in a common open affine neighbourhood. For (quasi)-projective varieties $X \subset \mathbb P^n$ this can never happen, since for any finite number of points $p_1, \dotsc, p_k \in \mathbb P^n$, there exists a hyperplane $H \subset \mathbb P^n$ which does not contain $p_1, \dotsc, p_k$, and $X \setminus H \subset \mathbb P^n \setminus H$ is a common affine neighbourhood.