Define the relation $\sim$ over $\mathcal C := \{0, 1\}^{\mathbb N}$ given by $(x_n)_n\sim (y_n)_n$ iff $\exists^\infty k: a_{k+i} = b_{k+i},~i=0, 1, \dots, k-1$.
What is the least size of a subset $\mathcal D\subset\mathcal C$ such that $\forall x\in\mathcal C~\exists y\in \mathcal D: x\sim y$?
In graph theory language, I'm looking for the domination number $\gamma(G)$ of the infinite graph $G = (\mathcal C, \sim)$.
Lemma. $\gamma(G)>\aleph_0$.
Proof. Let $S = \{s^m=(s_n^m)_n: m\in\mathbb N\}\subset \mathcal C$ be a countable subset. Then $S$ is not dominating. In fact, let $x = (x_n)_n\in\mathcal C$ be such that $x_{2^i+j}\ne s_{2^i+j}^j$, $0\le j<2^i$. Then $x\not\sim s^m$ since, given $n>m$, there is a number of the form $2^i+m$ in the set $\{n, n+1, \dots, 2n-1\}$. $\quad\square$
This lemma leave us in an awkward position. Of course $\gamma(G)\le 2^{\aleph_0} = |\mathcal C|$, which suggests $\gamma(G)$ might be a small cardinal, and the claim ''$\gamma(G)=2^{\aleph_0}$'' is consistent within ZFC. I wonder: is ''$\gamma(G)<2^{\aleph_0}$'' consistent within ZFC? One could prove this by finding a small cardinal $\mathfrak x$ such that $\gamma(G)\le\mathfrak x$.
I believe I can prove $\mathrm{cov}(\mathcal L)\le\gamma(G)$ by taking each $N(v)$, $v\in\mathcal C$, into a Lebesgue null subset of $\mathbb R$ through a surjection, but I didn't write the proof yet.
If $H$ is Cohen generic then $\mathbb R^V$ is dominating in the graph $(\mathcal C,\sim)$ as defined in $V[H]$: If $\dot x$ is a name for a function $x\colon\omega\rightarrow\omega$ then we can find $y\colon\omega\rightarrow \omega$ in $V$ so that $\dot x^H$ and $y$ agree infinitely often. Simply make sure that for each Cohen condition $p$ and every $n<\omega$, there is $q\leq p$ and $n\leq m$ so that $q\Vdash\dot x(\check m)=\check y(\check m)$. This is easily done as there are only countably many $p$ and $n$.
Now if $(z_n)_{n<\omega}\in V[G]$ is a node in $\mathcal C^{V[G]}$ and $x\colon\omega\rightarrow\omega$ is so that $x(k)$ codes the sequence $z_{2^k},\dots, z_{2^{k+1}-1}$ then if we find a real $y\in V$ which agrees with $x$ infinitely often, we can easily define some $(z'_n)_{n<\omega}\in V$ with $(z_n)_{n<\omega}\sim (z'_n)_{n<\omega}$.
We can now arrange $\gamma(G)=\omega_1$ with arbitrarily large continuum by forcing with a suitably long finite support product of Cohen forcing over a model of CH. Any real adjoined in this way belongs to a Cohen extension of $V$, so $\mathcal C^{V}$ will be dominating in the extension. This strongly suggests that $\gamma(G)\leq\mathrm{non}(\mathcal M)$ should be provable and indeed it is. If you want to prove that, I suggest to look at the notion of "matching a chopped real", see e.g.:
Blass, Andreas, Combinatorial cardinal characteristics of the continuum, Foreman, Matthew (ed.) et al., Handbook of set theory. In 3 volumes. Dordrecht: Springer (ISBN 978-1-4020-4843-2/hbk; 978-1-4020-5764-9/ebook). 395-489 (2010). ZBL1198.03058.
Keep us updated if you manage to pin $\gamma(G)$ down exactly with respect to all the other characteristics!