The question is simply postulating a certain model of the Data Generating Proces where the next value $y_0$ is obtained independently of the $n$ observations used to fit the model $y_i = x_i\beta + \epsilon_i$.
If the future observation $y_0$ is truly is obtained by independent randomly drawing from some population then the model is ex hypothesi right. The question is then how do you make sure you draw independently? And is $y_0$ really generated that way? It is in this case impossible to know - beyond the claim that this is the case - since the example is fictitious.
But that said, you should only apply this procedure if you find yourself in a setting where the value $y_0$ you are trying to predict is independent randomly generated. There are many settings where this will not be the case and many of these could be related to timeseries which is unfortunately somewhat implied by the use of the word "future", but I do not think you should read too much into that here.
EDIT: Also $x_0$ is treated as given, non-random.