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: 
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 $X\in[0,1]$, $x\in \mathbb{R}^+$.
In logistic regression binary variables may be standardise for combining them with continuos vars when you want to give to all of them a non informative prior such as N~(0,5) or Cauchy~(0,5).
The standardisation is adviced to be as follows:
Take the total count and give

1 = proportion of 1's

0 = 1 - proportion of 1's.

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Edit: Actually I was not right at all, it is not an standardisation but a shifting to be centered at 0 and differ by 1 in the lower and upper condition, lets say that a population is 30% with company A and 70% other, we can define centered "Company A" variable to take on the values -0.3 and 0.7.