I'm definitely not an expert in set theory, but Russel's paradox has long since been dealt with by making the class-set distinction.

$V$ is the *class* of all *sets*, not the class of all classes or the set of all sets, and this is really the whole shebang. We aren't allowed to collect 'all collections of the same nature' (sets or classes) into a collection of that same nature (one big set or class) on pain of paradox, but we can collect all collections of a certain nature (sets) into a new, bigger type of collection with a different nature (a class).

We can even continue this hierarchy with 'hyperclasses' that are allowed to hold all classes but not other hyperclasses, etc, as explained in the answer to [this MO question][1] by Joel Hamkins (and Andreas Blass/Kameryn Williams in the comments on Joels answer). 

In essence we can allow for a fundamentally 'bigger' type of collection, which can then hold all collections of a smaller type, but this new bigger type of collection will still never be able to collect up all collections of its own type --  we would have to once again step higher up the 'collection hierarchy', at which point we would run into the same situation again.


  [1]: https://mathoverflow.net/questions/292350/an-axiom-for-collecting-proper-classes