The guarantee that such a set can't exist is already given by the argument of Russell's paradox: its existence leads to a contradiction therefore it can't exist.
The problem with unrestricted comprehension was that it guaranteed the set does exist, which causes a problem because of the conflicting guarantees.
Yes, this is an issue.
Naively, this issue cannot be dealt with, and we'll get to that in a moment. But in 1917 mathematicians already noticed that "normal sets" do not contain themselves, and in fact have an even stronger property. Namely, there are no infinite decreasing chains in $\in$, so not only that $a\notin a$ it is also true that $a\notin b$ whenever $b\in a$, and that $a\notin c$ whenever for some $b\in a$ we have $c\in b$; and more generally there is no sequence $x_n$ such that $x_{n+1}\in x_n$ for all $n$.
This is exactly what the axiom of regularity came to formalize. It says that the membership relation is well-founded, which assuming the axiom of choice, is equivalent to saying that there are no decreasing chains. In particular $A\notin A$, for any set $A$.
But we know, nowadays, that it is consistent relative to the other axioms of modern set theory (read: $\sf ZFC$) that there are sets which include themselves, namely $x\in x$. We can even go as far as having $x=\{x\}$. You can even arrange for infinitely many sets of the form $x=\{x\}$.
This shows that naively we cannot prove nor disprove that sets which contain themselves exist. Because naive set theory has no formal axioms, and is usually taken as a subset of axioms which include very little from $\sf ZFC$ in terms of axioms, and certainly it does not include the axiom of regularity.
But it also tells us that we cannot point out at a set which includes itself, if we do not assume the axiom of regularity. Since these sets cannot be defined in a nontrivial way. They may exist and may not exist, depending on the universe of sets we are in. But we do know that in order to do naive set theory and even more, we can safely assume that this situation never occurs.
Best Answer
The issue here is the interpretation of the verb "contain". "Contain" here means contain as an element, not as a subset. By this interpretation, almost no natural sets one would think of contain themselves, and thus it's far from an impossibility. For instance, the empty set does not contain itself. It doesn't contain anything. Also the set $\{3\}$ does not contain itself. Its only element is $3.$ $\{3\}$ is not an element of it.
It's harder to think of a set that does contain itself. A non-rigorous example is "the set of all things that are not vanilla ice cream." Since "the set of all things that are not vanilla ice cream", whatever it is, is certainly not vanilla ice cream, this set is an element of itself.
And yet, per Russell, even though it seems a perfectly normal property for a set to not contain itself, the set of all sets that do not contain themselves is a paradoxical construct, since this set would contain itself if and only if it did not contain itself.