It seems to me that Russell's paradox rather is a "paradox" concerning relations.
Suppose we want to construct a graph (with finite or infinite number of nodes) and want some node to be adjacent to exactly those nodes that are not adjacent to them selves.
It's the same problem, which seems to arise from the fact that it is not possible to define relations with nodes of certain adjacent specifications.
And there are a lot of other examples of impossible constructions of relations.
Suppose we want to construct a graph and want some node $s$ to be adjacent to exactly those nodes $x$ such that:
- all chains $x\to x_1\to x_2\to x_3\to\cdots$ are finite;
- given a surjection $f$ for the construction, $f(x)\nrightarrow x$. $($Set $s=f(x)\dots)$
It seems to be necessary to point out that I don't mean that there is a paradox of Russell (it was just a paradoxical consequence of a construction), and that I don't know if mathematicians really mean that the construction of Russell say something about sets as such.
But I do believe that a lot of people think that Russell's paradox really is just about sets.