Where are we proving things? This matters a lot, because different choices of axioms for set theory will give different answers. According to the standard set of axioms - ZFC - both statements are true; however, in (say) NF, another set of axioms, they are false! So to show (in some system) that these aren't sets, we need to use some specific axioms in addition to Russell's paradox.
In ZFC, we use the axioms replacement and separation. Take for example the class $C$ of sets of cardinality 1. Then I can define a map from $C$ to the universe of all sets, sending $\{x\}$ to $x$. If $C$ were a set, then by replacement the image of $m$ - that is, the whole universe - would be a set. But separation, together with the argument of Russell's paradox, implies that the whole universe cannot be a set.
(Actually, replacement implies separation, so we only needed replacement, but whatever.)
Let me address the (non-)role of Foundation/Regularity here, based on the comments below.
Foundation sounds very relevant to questions of this sort, since Russell's paradox deals with self-containing sets. In particular, in many set theories without foundation (e.g. NF), there is a set-of-all-sets. So it is tempting to guess that, without foundation, we can't disprove the existence of a set of all sets.
However, it turns out that foundation is really a red herring here. It is easy to prove that there is no universal set in ZF minus foundation, as follows:
Suppose a universal set $U$ existed.
By Separation, we get the set $R=\{x: x\in U, x\not\in x\}$. Note that all we need here is that "$x\not\in x$" is a formula in the language of set theory - it doesn't matter what sort of formula it is, the Separation scheme applies to all formulas.
Now the usual Russell argument goes through. If $R\in R$, then $R\not\in R$; this is immediate (and more generally shows that for any set $A$, the set $R_A=\{X: X\in A, X\not\in X\}$ is not an element of $A$, although it is a set by Separation). More substantively, we know by assumption that $R\in U$ (since our assumption was that everything is in $U$); so if $R\not\in R$, that means $R\in U$ and $R\not\in R$, so $R\in R$.
So we have a contradiction; and this means that our initial assumption - that a universal set exists - was incorrect.
Note that this is a proof inside the theory ZF minus Foundation. I'm not arguing "If there is a universal set, then ZF minus Foundation is inconsistent;" I'm saying "ZF minus Foundation proves that there is no universal set." This proof inside ZF minus Foundation is a proof by contradiction, but that's fine.
