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.
What do you want to standardize, a binary random variable, or a proportion?

It makes no sense to standardize a binary random variable. A random variable is a function that assigns a real value to an event $Y:S\rightarrow \mathbb{R} $. In this case 0 for failure and 1 to success, i.e. $Y\in \lbrace 0,1\rbrace$.

In the case of a proportion, this is not a binary random variable, this is a continuous variable $X\in[0,1]$, $x\in \mathbb{R}^+$.