I think this is a very good and deep question that maybe related to Prof.Wen's [question](https://physics.stackexchange.com/questions/29311/what-is-spontaneous-symmetry-breaking-in-quantum-systems), and I would try to answer you in my understanding. 

Let's take the nearest-neighbor spin-1/2 antiferromagnetic Heisenberg Model on the square lattice as an example, where the symmetry breaking Neel state only emerges **in the thermodynamic limit**. 

As you have mentioned, note that only when the system has *finite size*, (i.e., the square lattice constitutes of two sublattices **A** and **B** with equal sizes $N_A=N_B$, and hence the total number of spins $N=N_A+N_B$ is **even**), the ground state is *unique* and is exact a *singlet* state with $SU(2)$ spin-rotation symmetry (Marshall,1955; Lieb and Mattis, 1962). However, as the system size becomes large, there are *many* low-lying excited states with very small energy gap $\Delta$ above the *singlet* ground state, and these low-lying states break the $SU(2)$ spin-rotation symmetry(i.e., they may be triplet states). More subtlely, as $N$ approaches $\infty $, those *nearly degenerate ground states* would 'collapse' into the ground state in the thermodynamic limit ($\Delta\rightarrow 0$), indicating that the Neel state is in fact a superposition of **many** *nearly degenerate ground states in the thermodynamic limit*. Thus, in the **strict** thermodynamic limit, there exists an $SU(2)$ symmetry breaking state of the 'highly' degenerate ground states.

Indeed, this is a nontrivial example of spontaneous symmetry breaking since the exact ground state of the finite system does not break $SU(2)$ spin-symmetry while there emerges spontaneous symmetry breaking (due to the nearly degenerate ground states) in the thermodynamic limit. The above argument is just a very rough picture, and I also feel it is somewhat difficult to understand how the ground state degeneracy happens for a gapless system in the thermodynamic limit? Moreover, I also get another question: Theoretically, as there are 'highly' degenerate ground states containing both $SU(2)$ symmetric singlet state and symmtery-breaking Neel states in the thermodynamic limit, why we are used to saying the ground state of the antiferromagnetic Heisenberg Model on the square lattice is a Neel state rather than a singlet state?