Is it possible that every set can be specified?

Yes.

J. D. Hamkins, D. Linetsky, and J. Reitz, Pointwise definable models of set theory, Journal of Symbolic Logic 78(1), pp. 139-156, 2013.

Perhaps slightly surprisingly, every pointwise definable model of ZF satisfies the Axiom of Choice (because if every set is definable, then in particular it is ordinal definable, and V=HOD implies AC). Intuitively one might think to look to the AC to find a candidate for an undefinable set, but actually it is the opposite that is the case: If we're looking at a situation where AC fails, then there must be an undefinable set somewhere.

The minimal model $M$ is such a model. $M$ is defined to be $L_{\delta}$ (the $\delta$-th level of constructible hierarchy) where $\delta$ is least such that $L_{\delta}$ models ZFC. Assuming that there are standard models of ZFC, the minimal model exists.