This was too long to be a comment, my apologies:

Further to Tim Gowers' comment and the following discussion, Chapter 6 of Bollobas' lovely white book "Combinatorics - set systems hypergraphs, families of vectors, and combinatorial probability" studies exactly this kind of question,.

In fact Tim's comment corresponds to Theorem 6 in that chapter of the book, I believe. The language of that theorem uses symmetric difference of subsets of $\{1,2,..,n\}$, which can be converted to inner products of normalized $n$ dimensional vectors with entries $\pm 1/\sqrt{n}$.

Let $\epsilon>0$ be small. The second case of Theorem 6 essentially states that if the normalized inner product is allowed to be in $[-1,\epsilon]$, equivalently, if the pairwise symmetric differences of the sets in the family are all slightly less than $n/2$ in relative terms, then the number of sets in the family, and hence the number of vectors with entries $\pm 1/\sqrt{n}$ can be as large as $2^{\epsilon n}.$

The proof of this part of the Theorem is given as an exercise with a nice hint in the book:
Focus on subsets of size $k=\lceil n/2 \rceil$ and do sphere packing in the Hamming space.

The variation with the OPs question is that the allowed range for the normalized inner product is $[-\epsilon, +\epsilon]$ in the OPs question. From Jelani Nelson's answer, it seems that the penalty for restricting the inner product to a band is perhaps not as severe as one might expect. We go from $\epsilon n$ down to $\epsilon^2 \log(1/\epsilon)n$ in the exponent.