There is work by Brumer and Kramer, *Paramodular abelian varieties of odd conductor*, on the possible conductors. They show for example that if $A$ is a semistable abelian surface over $\mathbb{Q}$ of odd non-square conductor $N$, then $N \geq 249$ (and there is an explicit abelian surface with this conductor). They also give tables of possible odd conductors $N \leq 1000$.

The Langlands philosophy predicts that abelian varieties of dimension $g$ over $\mathbb{Q}$ should (roughly) correspond to automorphic forms on $\mathrm{GSp}_{2g}/\mathbb{Q}$. There is even a precise conjecture, see Section 8 in the article of Brumer and Kramer. If you assume this conjecture, then you are lead to investigate the space of such automorphic forms, and for what levels the space is non-trivial. This gives only a necessary condition, because the field of Hecke eigenvalues may be larger than $\mathbb{Q}$. There has been work by Poor and Yuen, *Paramodular cusp forms* where they classify such automorphic forms. Their results support the above conjecture.