Anthony's answer settles the matter, but I'll say a few words about relevant terminology and references that are too long to fit in the comment box. (And go a bit beyond what you actually asked, but may give things some context.)

This is essentially a question in multifractal analysis. Given an asymptotic property such as "the asymptotic frequency of ones is bounded away from 1/2", one can study the set of points with this property in different ways. If you study the measure of this set, you're doing ergodic theory; if you study the dimension of this set, you're doing multifractal analysis.

In your setting, a natural thing to do is to fix $\alpha \in [0,1]$ and to consider the set $K_\alpha = \{ x \mid \frac 1n \sum_{k=1}^n x(k) \to \alpha \}$. (Each set $K_\alpha$ is contained in your non-recurrent set.) Then $2^{\mathbb{N}} = (\bigcup_{\alpha\in [0,1]} K_\alpha) \cup \hat K$, where $\hat K$ is the set of points $x$ for which $\lim \frac 1n \sum_{k=1}^n x(k)$ does not exist. This is an example of a *multifractal decomposition*. One can show that the measures $\mu_\alpha$ in Anthony's answer have the property that $\mu_\alpha(K_\alpha)=1$ and
$$
\dim_H K_\alpha = \dim_H \mu_\alpha = \frac{-\alpha\log \alpha - (1-\alpha)\log (1-\alpha)}{\log 2}.
$$
Thus the function $\alpha \mapsto K_\alpha$ is an analytic and concave function of $\alpha$; this is an example of a *multifractal spectrum*. There are lots of these, associated to various asymptotic quantities, and you can find a lot of subtle behaviour. (For example, one can ask how big the set $\hat K$ is, and it turns out that even though it is null for *every* shift-invariant measure on $2^\mathbb{N}$, it still has full Hausdorff dimension.)

For more on this, you can see the book "Dimension Theory in Dynamical Systems", by Yakov Pesin -- the first few chapters are very abstract and difficult to follow if you're not already pretty familiar with some of the basic ideas in dimension theory, but you can also skip straight to the chapters on multifractal analysis and get an idea of what's going on. There's also a survey paper by Barreira, Pesin, and Schmeling from 1997 or thereabouts that is a good introduction.