The idea is to consider the collection of all sets as another type of object.
Usually, such objects are called classes. Bernays-Gödel set theory is a (conservative extension of ZFC) theory that includes classes, and where therefore the class of all sets is a well-defined concept.
Clearly, the class of all classes would have the same problems as the set of all sets, but this is avoided by the fact that in BG it is not possible to quantify over classes.
Professional mathematicians that do not work in logic or set theory, i.e. most of them, take a more relaxed approach to classes, and mostly use them as semi-rigorous objects, being careful not to quantify over them but using them essentially as sets. One such example is category theory, where many of the categories commonly used are classes.