If I understand the question correctly, then maybe you are after special cases, as well as a general comment. So, as one of your examples suggests, one special case is to let G be a Banach space, considered as sitting inside its own bidual, and let $X=G^*$. Thus G induces the usual weak*-topology on X.
So an example of a positive answer is furnished by the Kaplansky Density Theorem: here G would be the predual of a von Neumann algebra, X would be a von Neumann algebra, and we let Y be any self-adjoint subalgebra which is weak*-dense. Then Kaplansky Density tells us that indeed the unit ball of Y is weak*-dense in the unit ball of X. This is an incredibly useful tool in Operator Algebra theory.
This then suggests that the result is unlikely to be true in general. Indeed, I think rpotrie's counter-example works! But here's an easier variant. Let $G=c_0$ and $X=\ell^1$, and let Y be the span of vectors $e_n+ne_{n+1}$. To see that this is weak*-dense, suppose that $\sum_k a_k e_k^* \in c_0$ annihilates all of Y. Then $a_n + na_{n+1}=0$ for all $n$, so $a_1 + a_2=0$ and $0 = a_2+2a_3 = 2a_3 - a_1$ and $0=a_3+3a_4 = (1/2)a_1+3a_4$, so $a_1 = -a_2 = a_3/2 = -a_4/3$ and an easy induction shows $a_1 = (-1)^{n-1}a_n/n$. Thus $|a_n| = n|a_1|$ for all $n$, but as $|a_n|\rightarrow 0$, it
follows that $a_1=0$, and so actually $a_n=0$ for all $n$. Hence Y is weak*-dense. However, $e_1$ is in the closed unit ball of X, but it's pretty clear that we can't approximate it by norm one elements in Y (to do this without a tedious calculation defeats me right now).
Let me try a possible answer. Take a model of $ZF$ where the axiom of choice for a denumerable family of finite sets holds but where there is an infinite Dedekind finite set $B$ (this model can be checked to exist, for instance, here; $\mathcal{M}32$ is one such model). Then $\ell_2(B)$ is an infinite dimensional Hilbert space with a Dedekind finite orthonormal base, whose unit ball is, by theorem 2 of the previously cited article, sequentially compact.
Best Answer
It's exercise V.1.10 in J. Conway, A Course in Functional Analysis, 2e, if that's any help.