Another common way do describe the error is the following:
Suppose you have some variable $Y$ which you model as a linear function of another variable $X$, that is, you want to see how well the function $f(X):=\alpha+\beta X$ describes $Y$ according to some measure. Then define $\epsilon$ as the difference between $f(X)$ and $Y$. Then $\epsilon$ is all things affecting $Y$ not captured by $f(X)$. To me, this is the most basic and intuitive way to think about the error term.
It's everything in $Y$ not explained by a linear (affine) function of $X$.
For example, if you let $Y=$ average age of citizens in a country, and then you model this as an affine function of how much they exercise and how healthy they eat ($X$="Healthy Lifestyle Index"), then $\epsilon$ is everything affecting how long people live on average except for an affine function of some healthy lifestyle index. This includes effects from smoking, the health care system, happiness, climate, etc.