"Give him a box that everyone knows what it contains."

"Give him a box that everyone knows what it contains."

Is this correct English? It sounds wrong to me.

Congratulations, @Eric, you've just rediscovered one of the Ross Constraints.
These are constraints on syntactic rules of certain kinds that were discovered
by Haj Ross, in his 1967 MIT dissertation Constraints on Variables in Syntax.

With the sentence you provide

• Give him [a boxᵢ [that everyone knows [what itᵢ contains]]]

There are three nested clauses. One, Give him a box, is the main clause.
There are two relative clauses, one modifying the other:

• the free relative (or embedded question) what it contains, from the things that it contains.
  The things that it contains, in turn, is the direct object of the verb knows in
• the relative clause that everybody knows what it contains,
  which modifies the noun phrase a box, the direct object of the main clause.

The problem is that, according to the rules of relative clause formation, that it shouldn't be there. Suppose the sentence was

• *Give him a book that everybody has read it.

That sentence is ungrammatical (that's what the asterisk means).
If you're a native English speaker, you can tell by the feeling you get.
The relative clause rules should move it to the front, and change it to which, or that, producing

• Give him a book that/which everybody has read.

which is normal. But that didn't happen in the original sentence.
To see why, compare the original sentence with and without the it

• *Give him a box that everyone knows what it contains.
• *Give him a box that everyone knows what contains.

Neither one is terrific, but the second one is just godawful,
while the first one, by comparison, seems only mildly off.

What Ross discovered was that there were constructions, which he called "islands" -- headless relative clauses are one kind -- which set up boundaries to movement rules like the movement part of relative clause formation.

In other words, that it can't just be yanked out and moved to a position after box,
because it's inside a headless relative, which is an island and whose contents are therefore protected against extraction rules.

This post goes into more detail about syntactic islands; Part 5, at the end,
explains why this kind of sentence occurs, using a different example pair:

• *That's the book that Bill married the woman who illustrated it.
• *That's the book that Bill married the woman who illustrated.

"Give him a box" and "everyone knows what it contains" are both fine as independent units. The problem is with the attempt to splice them together with that. This is because everyone knows what it contains ought to be a dependent clause, but is constructed like a main clause.

The solution is to reword the second element so that it becomes a true dependent clause. For instance:

"Give him a box whose contents everyone knows."

"Give him a box which everyone knows the contents of."