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][1], 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][2]. 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][3] 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 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 "Tom said that a teddy bear from 1996 is not an antique." The speaker is just using the *content clause* from this sentence (see above). You're allowed to do this in conversation when the immediate dialogue makes it clear how to finish your sentence.

It's very similar to the following dialogue

> A: Who broke the window?

> B. Maria.

Here, 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.




  [1]: https://en.wikipedia.org/wiki/Direct_speech
  [2]: https://en.wikipedia.org/wiki/Indirect_speech
  [3]: https://en.wikipedia.org/wiki/Content_clause