This question is not answerable. First, we would need “everything” to be well-defined, and arguably, to be useful, the definition would not be self-referential. This being possible while decently approximately preserving our intuition of “everything” is questionable. Secondly, for canonical and the most useful approaches to logic, the idea requires quantification over everything, which is problematic even if “everything” is well-defined. In order for quantification to make sense canonically, it must be over a domain of discourse or your system must have restricted comprehension.
Not having a domain of discourse or having unrestricted comprehension leads to many paradoxes, such as Russell’s paradox and Curry’s paradox. The former is more famous, but I’d argue the latter is more relevant to your question. Once you specify a domain of discourse, there will be something outside of it, and restricted comprehension is precisely that – restricted, so not all-inclusive.
To see an example of what can go wrong, does your domain of discourse properly include itself? If it doesn’t, then it doesn’t include everything. If it does, then if we are speaking of sets, your system will fall prey to Russell’s paradox and will be inconsistent.
Does your system quantify over arbitrary formulas or can formulas be instantiated by variables? If your system does not, then you can’t give semantic definitions to all the formulas inside your system (they can still be well-formed, structurally speaking). If your system does, then your system likely falls prey to Curry’s paradox.
A specific, formal example of something that cannot be defined within the standard model of a system that can interpret arithmetic is “truth”. This is Tarski’s Undefinability Theorem. Informally, any consistent system that can interpret arithmetic cannot define its own truth predicate. There is an even more general form of the theorem with a wider scope.
What if we instead try to work with a natural language instead of a formal system? Not only will you still encounter Curry’s paradox, but you’ll have to deal with Richard’s Paradox as well…
Thus the resolution of Richard's paradox is that there is not any way to unambiguously determine exactly which English sentences are definitions of real numbers
Very informally speaking, to talk about everything, you want your system to be completely open (to include everything) while still being closed (in order to give rules, be manageable, and be consistent). This is not possible. The more inclusive your system is, the more you can speak about but also the more likely your system contains a contradiction. An intuitive analog to this is a bag which contains everything is impossible because everything must be inside the bag, but if there’s an inside then there must be an outside – there’s no such thing as “inside” without “outside” as well. Also, is such a bag properly inside itself?
So why isn’t the answer to your question “no”? Because these paradoxes emerged given the way we normally approach logic. There may be some other categorically distinct approaches we don’t know about. Because we don’t know them and because “everything” is not well-defined, your question is fundamentally unanswerable.