Is every abelian variety a subvariety of a Jacobian?

Embed the dual abelian variety into projective space. Take a smooth hyperplane section and interate until it's one-dimensional, obtaining a smooth curve $C$. By Lefschetz $C$ is irreducible, and the natural map $H_1(C, \mathbb Z) \to H_1(A^\vee, \mathbb Z)$ is surjective. Because $H_1(C, \mathbb Z)= H_1(J(C), \mathbb Z)$, the natural map $H_1(J(C), \mathbb Z) \to H_1(A^\vee, \mathbb Z)$ is surjective as well. Now this does imply that the map of dual abelian varieties is surjective because abelian varieties over $\mathbb C$ are Pontryagin dual to the integral homology of the dual abelian varieties.

In characteristic $p$, a similar argument should work using the injectivitity on etale cohomology with torsion coefficients and crystalline cohomology with torsion coefficients / algebraic de Rham cohomology, but I didn't check the details.

Let me give an answer for $k = \mathbb{C}$.

By a theorem of Matsusaka, every abelian variety $A$ over an algebraic closed field $k$ is a quotient of a Jacobian.

Now just apply Matsusaka's theorem to $A^{\vee}$, and dualize. Since we are over $\mathbb{C}$, dualization sends surjective morphisms of Abelian varieties into injective ones, so we are done.

I think that the slightly weaker version where the field is infinite is proved somewhere in Milne's lecture notes on Abelian Varieties.

Reference.

T. Matsusaka: On a generating curve of an Abelian variety, Nat. Sci. Rep. Ochanomizu Univ. 3 1-4, (1952).

Quoting from P. Samuel review on MathSciNet:

An abelian variety $A$ is said to be generated by a variety $V$ (and a mapping $f$ of $V$ into $A$) if $A$ is the group generated by $f(V)$. It is proved that every abelian variety $A$ may be generated by a curve defined over the algebraic closure of $\mathrm{def}(A)$.

You can find a detailed proof here (theorem 1.2) in the case of principally polarized abelian varieties. One reduces to this case using the Zarhin's trick. The assumption of $k$ being infinite should not be necessary (see remark 1.3 in the paper)