To attempt an answer to your original question, each $\epsilon_i$ is considered to be one realization of the random variable $\epsilon$. So each $\epsilon_i$, $i = 1,\ldots,n$, is indeed only one number, but there is an underlying distribution which generates those numbers. I have never seen the notion of homo/heteroscedasticity as referring to any single one $\epsilon_i$ in particular (and would certainly not choose that phrasing myself), rather the concept refers to the whole collection of error terms.
A simple example would be to consider that you are taking 50 measurements manually, using some tool that requires hand stability, so your sample is $y_1,\ldots,y_{50}$. Then a friend of yours with shaky hands takes another 50 measurements, $y_{51},\ldots,y_{100}$. Intuitively, if you model your 50 observations separately from those of your friend, you'd expect that your $\epsilon_i, i=1,\ldots,50,$ would tend to deviate less from their mean value than those $\epsilon_i, i=51,\ldots,100$ your friend would get from his model on the same predictors, because you know you have steadier hands. This means that your $\epsilon_i$ are in fact homoscedastic within their own group, i.e. for $i=1,\ldots,50$ (assuming you don't get tired after a while). What's more, your friend's errors are also homoscedastic with respect to his group, i.e. $i=51,\ldots,100$. If you pool everything together though, the collection $\epsilon_i, i=1,\ldots,100$ is decidedly not homoscedastic, because not all of these $\epsilon_i$ were generated by the same mechanism (half the errors came from your hands, the other half from your shaky friend).
Another example of heteroscedasticity would be to take all 100 measurements yourself but (much more realistically) assume fatigue with time: you'd see that your $\epsilon_i$ would be very low for the measurements you took first, but they would increase with time as your muscles grow tired and your precision suffers, thus increasing the variance of future errors (i.e. the chance that you make bigger errors in the measurement). Hope this helps.