Here a relative clause construction (“Den” rather than “Ihn”) stands as a complete sentence, which I imagine might be something like “Wir alle wussten, dass er ein Mann war, den niemand einstellen würde.”
Common? Literary? Old fashioned?
This is not a relative clause, and den is not a relative pronoun in this case. If it was, you could replace it with “welchen”, which wouldn’t work here.
It is rather a demonstrative pronoun, or definite article hybrid. It is equivalent to “Diesen” in this sentence.
What is missing here is a noun after it:
Den (Mann/Typen/etc.) würde niemand einstellen.
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Let me refine my answer.
What we have here is a definite article that is used as and functions as a demonstrative pronoun.
I found a webpage that explains this:
German Demonstrative Pronouns | Study.com
Der , die and das (pronounced dehr, dee, dahs) are the German definite articles, but they will often be used as demonstrative pronouns as well. They are used in situations when the person or thing has been mentioned previously, and we know exactly what it’s referring to and are mostly translated as ‘her’, ‘him’ or 'it.
This is not the first time this comes up in this forum. We’ve just had a similar question a while ago:
Haben die es schon der Polizei gemeldet? - #2 by pitti42
In this sentence it is not “der” because it’s actually a dative object (niemand is the subject), so as you see it’s not just the nominative.
And another thing: It does not have to be something previously mentioned. Pointing at someone or something while speaking works as well. Or a nod with the chin 
Often “da” is added when pointing:
Den da würde niemand einstellen.
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