For the benefit of others who read this, note that if $\mathcal H$ is a separable Hilbert space, then there exists a countable, orthonormal basis for $\mathcal H$. Notice that this does not immediately imply that there cannot exist an uncountable basis for $\mathcal H$, but this nonetheless turns out to be true as a consequence of the dimension theorem;
Let $V$ be a vector space, then any two bases for $V$ have the same cardinality.
See also the following math.SE posts:
http://math.stackexchange.com/questions/232166/showing-the-basis-of-a-hilbert-space-have-the-same-cardinality
http://math.stackexchange.com/questions/450106/uncountable-basis-and-separability
Now, for your question. Let's suppose that there are an uncountable number of orthogonal vacuua $|\theta\rangle$ in the Hilbert space where $\theta\in [0,2\pi)$, then we have the following possibilities
The Hilbert space of the theory is not separable. In this case, there is no contradiction.
The Hilbert space of the theory is separable. In this case, there is a contradiction, and we need a resolution.
As far as I am aware, most axiomatizations of QFT assume that the Hilbert space of the theory is separable, but there is discussion in the literature about relaxing this assumption. I'll attempt to dig up some references.
Let's therefore assume separability and look for a resolution. The standard resolution is that when constructing the Hilbert space of the theory, one chooses only one of these (physically equivalent) vacuua to be the vacuum of the Hilbert space, then one constructs the rest of the physical Hilbert space about this vacuum. The rest of the vacua are not elements of the Hilbert space of the theory.
There is another perspective on this which is interesting. Let's suppose that there is some larger, non-separable Hilbert space $\mathcal H_\mathrm{big}$ containing all of the vacua $|\theta\rangle$ and which is an orthogonal direct sum all of the Hilbert spaces $\mathcal H_\theta$ that could have been generated from each of the possible vacua and used as the physical Hilbert space of the theory.
\begin{align}
\mathcal H_\mathrm{big} = \bigoplus_{\theta\in[0,2\pi)} \mathcal H_\theta
\end{align}
Then we view each of the Hilbert spaces $\mathcal H_\theta$ as a superselection sector of the larger Hilbert space $\mathcal H_\mathrm{big}$. In this case, if the physical system occupies a state $|\psi\rangle$ in a given superselection sector $\mathcal H_\theta$, then the state of the system will remain in the sector for all times under the Hamiltonian evolution, so we may as well view "the" Hilbert space of the system as simply the superselection sector it started in. In a sense, this is essentially the same as originally having picked a vacuum on which to build the Hilbert space because the different superselection sectors don't "talk" to each other.
The following physics.SE post is useful for understanding superselection sectors:
What really are superselection sectors and what are they used for?
I also found the following nLab page on superselection theory to be illuminating:
http://ncatlab.org/nlab/show/superselection+theory
This post imported from StackExchange Physics at 2014-03-31 16:08 (UCT), posted by SE-user joshphysics
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