A related statement is known in all dimensions: Boucksom, Demailly, Păun and Peternell showed that a smooth projective variety X over a field of characteristic zero is uniruled if and only if the canonical bundle of X is not pseudo-effective (that is, not in the closed convex cone spanned by effective divisors in the Néron-Severi group tensored with the real numbers).
[1] As a very special case, a smooth hypersurface of degree d in Pn over a field of characteristic zero is uniruled if and only if d ≤ n, by the adjunction formula.
(In fact, a smooth hypersurface of degree d ≤ n in Pn is a Fano variety and hence is rationally connected, which is stronger than being uniruled.)
A variety X over an uncountable algebraically closed field k is uniruled if and only if there is a rational curve passing through every k-point of X.
By contrast, there are varieties over the algebraic closure k of a finite field which are not uniruled but have a rational curve through every k-point.
(The Kummer variety of any non-supersingular abelian surface over Fp with p odd has these properties.
[2]) It is not known whether varieties with these properties exist over the algebraic closure of the rational numbers.
In the positive direction, every uniruled variety of dimension at most 2 over an algebraically closed field of characteristic zero is ruled.
("Separable" means that the derivative is surjective at some point; this would be automatic for a dominant rational map in characteristic zero.)