Let me begin quoting W. Tait (lectures on proof theory, pages 4 and 5):

> I believe that what further has to be understood, in order to make
sense of these 'paradoxes' is that the notion of a transfinite number or, equivalently,
of a set of transfinite numbers is an essentially **open-ended** notion:
no matter what principles we introduce to construct sets of numbers, providing
only that these principles are well-defined, we should be able to admit all
numbers obtained by these principles as forming a set, and then proceed on
to construct new numbers. So $\Omega$ cannot be regarded as a well-defined extension:
**we can only reason about it intensionally**, in terms of those principles
for constructing numbers that we have already admitted, leaving open in our reasoning the possibility - in fact, the necessity - of always new principles for
constructing numbers. When this is not understood and $\Omega$
is counted as a
domain in the sense of a well-defined extension, then the so-called paradoxes
force on us a partitioning of well-defined extensions into two categories: sets
and proper classes; and the only explanation of why such an extension should
be a proper class rather than a set would seem to be simply that the assumption
that it is a set leads to contradiction. The paradoxes deserve the name
'paradox' on this view because it offers no explanation of why there should
be this dichotomy of well-defined extensions.


Emphases are mine ($\Omega$ is a reference to "the greatest ordinal"). So, one proposed rough sketch of an answer in the direction given by Tait (of course, there are many other directions in philosophy) is this: 

- The subject matter of set theory is open-ended, therefore set theory must be about an intension, the concept of set, not about a well-defined extension. This intension is open-ended (it is hard to make sense of the oxymoron "open-ended well-defined extension"), and it is the unifying criterion behind the plurality of set theoretical practices. The contemporary criterion can be more or less approximated by $ZFC$, but there can be no definite final stage on the progressive conceptual unification of the set-theoretical practices, as there is a neccessary open-endedness (incompleteness) in this intension.

There are many things to address here, but I will not try to pursue them, not even in outline, as this would lead us to a more hardcore philosophical activity.
As a final remark, there are similar arguments in the history of philosophy which were given many years before Russell. One of the most relevant is Plato's third man argument, in *Parmenides*.