"They know not of what they speak."

Solution 1:

Despite the word order being unusual by today's standard, the phrase is grammatically correct. To convert it to a more modern variant, move the not ahead of the verb (which requires adding do as well) and if you like, move the preposition to the end:

They don't know what they speak of

I'll note the usage of not after the verb is not uncommon in old phrases. E.g.:

Judge not, lest ye be judged

Solution 2:

I have to disagree with Dusty's previous explanation.

The original phrase, “they know not of what they speak,” is correct, but the explanation is different. Correctly rephrased, it means:

They do not know anything about (the thing) they are talking about.

Dusty's rephrase says, they don't know what they speak of and this, naturally, implies something very different. It may mean, for example, that they are drunk and have no idea whatsoever that they are even talking :)

In the original phrase, the speaker (described in the phrase), is fully confident and aware of what they are speaing.

Solution 3:

"Naught" is possible there, but it is not necessary to emend it: as others have said "they know not" is perfectly good English up to a couple of hundred years ago, and still not quite dead.

(Historically, "not" actually does come from "naught". The sequence (in Old English, but using modern forms of words) goes

"They ne know" -> "they ne know naught" -> "they ne know not" -> "they know not"

and then in Modern English

"They do not know"

See http://en.wikipedia.org/wiki/Jespersen%27s_Cycle)

Solution 4:

I think that 'know not' means 'do not know' and 'naught' means 'nothing'. So it would be like saying 'They do not know / they know nothing.'