Before I answer your question, we need to review the difference between direct speech and indirect speech.
Quotation marks are used for reports of direct speech, that is, reports which attribute specific words to someone. For example,
Tom: Flies like honey.
Mike: Tom said "Flies like honey." [Direct speech]
Quotation marks are not used for indirect speech. Further, using a 'that' is an optional way to introduce the clause that paraphrases what the person being quoted said. These are called content clauses or that-clauses. This use of 'that' is called the complementizer. Here is an example:
Tom: Flies like honey.
Mike: Tom said (that) flies like honey. [Indirect speech]
Notice that Mike is allowed to say 'that' here.
Now, on to your question:
"That a teddy bear from 1996 is not an antique" is NOT a grammatical sentence by itself.
When speaker A says "That a teddy bear from 1996 is not an antique", he is not using a full sentence. Rather, he is using an elliptical sentence. His full sentence would be the indirect speech report: "The officer said that a teddy bear from 1996 is not an antique." Speaker A is just using the content clause from this sentence (see above). You're allowed to do this in conversation when the immediately preceding context makes it clear how to finish your sentence.
It's very similar to the following dialogue:
A: Who broke the window?
B: Maria.
Here, speaker B is not using a full sentence. But context makes it clear that she means something like "Maria broke the window". The same thing is happening in your case.
