If I understood the question correctly, what you're asking is related to the maximum distance of binary codes with large minimum distance $d\geq s$.

In coding theory, $A_q(n,d)$ is defined as the maximum cardinality of a $q-$ary code with length $n$ and minimum distance $d.$

You can never have more than 2 codewords at full distance by binary geometry. In fact,
the Plotkin bound for high distance binary codes states:

If $d$ is even (thus $n$ even for your case) and $2d>n\geq d,$ then
$$
A_2(n,d)\leq 2\left\lfloor \frac{d}{2d-n}\right\rfloor,
$$
which will give $A_2(n,d)\leq 2,$ for your case. Take any vector and it's bitwise complement.

The $d$ odd case is similar, see for Example Roman's book on Coding and Information Theory.

Of course you only want a lim sup tending to 1, and **you can get it to tend to $1/2$** by using the rows of Hadamard matrices for $n$ a power of 2, but I doubt that any value larger than $1/2$ in your expression is achievable (see update below).

**Edit:** Thanks to Aaron Meyerowitz for clarifying the finite odd length case.

**Proposition:** The lim sup is actually $1/2.$

Assume that a value $d$ larger than $n/2$ is achievable. Map the codewords to $\pm 1$ vectors by writing $((-1)^{x_1},\ldots,(-1)^{x_n}).$ The inner product pf two $\pm 1$ valued vectors at hamming distance $d$ from each other is $\delta=n-2d.$ Therefore, if a collection of $m$ distinct $\pm 1$ vectors have minimum distance $d,$ we can write
$$
0\leq \left| \sum_{i=1}^m u_i \right|^2 = \langle \sum_i u_i , \sum_i u_i\rangle = \sum_i |u_i|^2 + 2 \sum_{i<j} \langle u_i,u_j \rangle \leq mn + m(m-1) (n-2d),
$$
which eventually yields
$$
d\leq \frac{n}{2}\frac{m}{(m-1)}.
$$
Letting $m=n$ proves the claim.