Must a word be binary (and never unary)?
Machine words have fixed length. Denoting the length by $n$ and the base by $b$, each machine word has $b^n$ possible values, and so stores $\log_2 (b^n)$ bits of information.
A base can be binary or decimal and so forth; if $b=1$ then the machine word has a single possible value, and so stores no information. It would be useless.
Unary encoding is only relevant in contexts where the length is not fixed.