Question: I have a data set with a set of features. Some of them are binary $(1=$ active or fired, $0=$ inactive or dormant), and the rest are real valued, e.g. $4564.342$.

I want to feed this data to a machine learning algorithm, so I $z$-score all the real-valued features. I get them between ranges $3$ and $-2$ approximately. Now the binary values are also $z$-scored, therefore the zeros become $-0.222$ and the ones become $0.5555$.

**Does standardising binary variables like this make sense?**

Answer: 
Standardizing binary variables does not make any sense. The values are arbitrary; they don't mean anything in and of themselves. There may be a rationale for choosing some values like 0 & 1, with respect to numerical stability issues, but that's it.
One nice example where it can be useful to standardize in a slightly different way is given in section 4.2
of Gelman and Hill (<http://www.stat.columbia.edu/~gelman/arm/>). This is mostly when the interpretation of the coefficients is of interest, and perhaps when there are not many predictors.

There, they standardize a binary variable (with equal proportion of 0 and 1) by
$$
\frac{x-\mu\_x}{2\sigma\_x},
$$
instead of the normal $\sigma$. Then these standardized coefficients take on values $\pm 0.5 $ and then the coefficients reflect comparisons between $x=0$ and $x=1$ directly. If scaled by $\sigma$ instead then the coefficient would correspond to half the difference between the possible values of $x$.