We can compare the issue regarding the definition of set with Geometry.
Euclid's Elements opens with five definitions :
A point is that which has no part.
A line is breadthless length. [...]
A surface is that which has length and breadth only.
They can be of some help in grasping the basic concepts, but hardly they can be conceived as real definitions at all.
In 1899 David Hilbert's published his groundbraking book on the axiomatization of geometry : Grundlagen der Geometrie, based on previous lectures. These are the first paragraphs (page 3):
Let us consider three distinct systems of things. The things composing the first system, we will call points and designate them by the letters A, B, C,...; those of the second, we will call straight lines and designate them by the letters a, b, c,...; and those of the third system, we will call planes and designate them by the Greek letters alpha, beta, gamma. [...]
We think of these points, straight lines, and planes as having certain mutual relations, which we indicate by means of such words as “are situated,” “between,” “parallel,” “congruent,” “continuous,” etc. The complete and exact description of these relations follows as a consequence of the axioms of geometry.
Hilbert's work on foundations of mathematics and logic has been called Formalism and it is still the prevailing philosophical view between "working" mathematicians.
For set we can consider Georg Cantor's mature definition of set in "Beiträge zur Begründung der transfiniten Mengenlehre", Mathematische Annalen (1895-97, Engl.transl.1915 - Dover reprint), §1, page 85 :
By an "aggregate" (Menge) we are to understand any collection into a whole (Zusammenfassung su einem Ganzen) M of definite and separate objects m of our intuition or our thought. These objects are called the "elements" of M.
Compare it with a modern textbook on set theory : Nicolas Bourbaki, Elements of Mathematics : Theory of sets (1968 - 1st French ed : 1939-57), page 65 :
From a "naive" point of view, many mathematical entities can be considered as collections or "sets" of objects. We do not seek to formalize this notion, and in the formalistic interpretation of what follows, the word "set" is to be considered as strictly synonymous with "term". In particular, phrases such as "let $X$ be a set" are, in principle, quite superfluous, since every letter is a term. Such phrases are introduced only to assist the intuitive interpretation of the text.
Thus, from a mathematical perspective, points and lines are "things" satisfying the axioms of geometry; in the same way, sets are "objects" satisfying the axioms of set theory.
Of course, also if a definition "inside" set theory of the notion of set is impossible, we can still have attempts to elucidate the notion of set from a philosophical perspective.
See e.g. Paul Benacerraf & Hilary Putnam (editors), Philosophy of Mathematics: Selected Readings, (2nd ed : 1983), Part IV. The concept of set.