Can the strange requirement that direct products exist only for small indexing families be relaxed, saying that all products (or limits) exists but some are outside of our category (and possibly belong to an other category)?
This sounds like that the roots exist for all real numbers but some should be considered in a greater set (complex numbers).
Can something like this be invented for categories?
The technology of Grothendieck universes handles many instances of this question; indeed, one might say it's designed to handle this and similar questions where size considerations interfere with what one might want to do. Specifically, suppose we fix a Grothendieck universe $U$ and take "small" to mean "member of $U$". Then in many cases, the large-indexed products you want will exist in a category, analogous to the one you started with, but in a larger Grothendieck universe $U'$. Here "analogous" is an imprecise term, but usually it can be taken to mean that one uses in $U'$ the same formal set-theoretic definition which, applied in $U$, defined the original category.