Let $\mathbf{1}$ be the all-ones vector, and suppose $\mathbf{1}, \mathbf{v_1}, \mathbf{v_2}, \ldots, \mathbf{v_{n-1}} \in \{-1,0,1\}^n$ are mutually orthogonal vectors. Does it follow that $n \in \{1,2\} \cup \{4k : k \in \mathbb{N}\}$?
A few notes are in order:
If we do not allow "$0$" entries in the vectors then this is a well-known necessary condition for the existence of an $n \times n$ Hadamard matrix, so the answer to my question is "yes" if we replace $\{-1,0,1\}$ with$\{-1,1\}$.
With some computer help, I have shown that there do not exist collections of vectors with these properties when $n = 3$, $5$, or $6$, so the answer to my question is "yes" when $n \leq 6$.