No, not even $\mathsf{DC}$ suffices for this. Here, $\mathsf{DC}$ is the axiom of dependent choice, which is strictly stronger than countable choice.
For instance, it is a theorem of $\mathsf{ZF}$ that for any set $X$, the set $\mathcal{WO}(X)$ of subsets of $X$ that are well-orderable has size strictly larger than the size of $X$. This is a result of Tarski.
Now, it is consistent with $\mathsf{ZF}+\mathsf{DC}$ that $\omega_1\not\le|\mathbb R|$, that is, there is no injection of $\omega_1$ into $\mathbb R$, and therefore $[\mathbb R]^{\aleph_0}=\mathcal{WO}(\mathbb R)$.
For a proof of Tarski's result, see for instance
MR1954736 (2003m:03076). Kanamori, A.; Pincus, D. Does GCH imply AC locally? In Paul Erdős and his mathematics, II (Budapest, 1999), 413–426,
Bolyai Soc. Math. Stud., 11, János Bolyai Math. Soc., Budapest, 2002.
(A really nice paper that everyone should read, by the way.)
For the remark regarding the consistency with $\mathsf{DC}$ of $\omega_1$ not injecting into $\mathbb R$, this holds for instance in Solovay's model. Details can be found in several places, such as Section 11 of
MR1321144 (96k:03125). Kanamori, Akihiro. The higher infinite.
Large cardinals in set theory from their beginnings. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1994. xxiv+536 pp. ISBN: 3-540-57071-3.
This is admittedly a rather curious result: It is provable in $\mathsf{ZF}$ that $\mathbb R$ and $\mathbb R^\omega$ have the same size, and there is a natural surjection from $\mathbb R^\omega$ onto $[\mathbb R]^{\aleph_0}$. This provides us with an example of a quotient of size strictly larger than that of the set it comes from.