What's going on in the example you quote is different from the constant-function issue. We don't actually mean that $s$ is either the constant function $0$ or the constant function $1$. We mean $s$ to be a proper random variable, we just haven't defined it fully.
Random variables without a sample space
Formally, a random variable needs a sample space to be defined: a random variable $\mathbf X$ is a function from the sample space $S$ to some set $R_{\mathbf X}$, the range of $\mathbf X$.
In practice, it's often true that whatever we want to know about $\mathbf X$ only depends on its distribution, and not on whatever is going on in the sample space. In the case of a discrete random variable, this means that once we specify $\Pr[\mathbf X = x]$ for every $x \in R_{\mathbf X}$, we know "everything we need to know" about $\mathbf X$, and don't need to say what $\mathbf X$ is as a function, or even what the sample space is.
If this makes you uncomfortable, you can always take the most boring random experiment that $\mathbf X$ could possibly live in. Let the sample space $S$ be $R_{\mathbf X}$ and define $\mathbf X$ to be the identity function; then, define $\Pr[A]$ for every event $A \subseteq S$ to be whatever you want $\Pr[\mathbf X \in A]$ to be.
In your example, we can define our sample space $S = \{0,1\}$, with $\Pr[\varnothing] = 0$, $\Pr[\{0\}] = \Pr[\{1\}] = 0.5$, and $\Pr[\{0,1\}] = 1$. Then, let $s$ be identity function on $\{0,1\}$. That's the random variable being described.
Of course, a random variable with the same distribution could be defined on other sample spaces. But if someone is talking about a random variable in this "sloppy" way, it's a good bet that this won't make a difference.
Random variables that are constant
Whichever approach we take to defining random variables, there's nothing wrong with a random variable $\mathbf X$ that's the constant function. For example, you could define the random variable $\mathbf 5$ as a function mapping everything in the sample space to $5$. Then the distribution of $\mathbf 5$ is that it's equal to $5$ with probability $1$.
We usually don't really care to do this for its own sake, but sometimes this happens accidentally. For example, the conditional expectation $\mathbb E[\mathbf X \mid \mathbf Y]$ is defined to be the random variable that's equal to $\mathbb E[\mathbf X \mid \mathbf Y=y]$ whenever $\mathbf Y=y$. If $\mathbf X$ and $\mathbf Y$ happen to be independent, then we've ended up defining a random variable that's always equal to the constant $\mathbb E[\mathbf X]$.
I'd rather distinguish the constant $5$ from the random variable $\mathbf 5$, but I can't think of a case in which it would matter.