We have hundreds of sterilite shoebox bins from Walmart ($1pc) and we use 2 week pill containers ($0.50pc) for screws when we do tear downs. We also have about a dozen bigger bins ($3-5pc) for tear downs of bigger laptops (gaming laptops). Try to stick with only a few types/sizes so they can easily be stacked on a shelf when not in use.
EDIT: As you can see, we not only use them for tear downs, but we use them to keep just about everything organized.