In the real world, these types of problems primarily happen in manufacturing. You usually see them when there are strong constraints on the behavior of a variable. For example, the normal distribution assumes that a value can take on any value over $(-\infty,\infty)$ but if you are building cars, it is never going to happen that a tire will be larger than the factory it is being constructed in, let alone of nearly infinite size. Indeed, barring a monumental equipment failure, the diameter will never be much outside some easily and well-defined maximum or minimum. There also will never be a penny-sized tire either. I am not sure what a negative diameter could mean.
A simple example would be a submarine hiding in the ocean. Because it is of fixed size and shape its variance is fixed. Its location is not fixed. Indeed, it is hiding.
You might have some way to collect data about the submarine's location. The data could be a location somewhere on the ship, for example, a point near the fantail. Maybe it could be from a variety of points around the vessel. If the data collected depends on the geometry of the ship, then the data generation function will have a fixed variance. However, as the mean is somewhere in the ocean we do not know what it is.
One other note, not all formulations of the variance assume that the mean is known or that a point estimate exists for it. Consider the posterior probability $\Pr(\mu;\sigma^2|X)$ where $X$ is the observed data, $\mu$ is the population mean, and $\sigma^2$ is the population variance. A point estimate of the variance can be obtained by first marginalizing out $\mu$ so that $$\Pr(\sigma^2|X)=\int_{-\infty}^{\infty}\Pr(\mu;\sigma^2|X)\mathrm{d}\mu.$$
From that point, a utility function could be imposed upon the distribution and a point found. While there is information about $\mu$, it is a probability distribution instead of a known point or an estimator. There is more than one way to estimate variance depending on the goals and the circumstances.