Is there a sentence that begins with “them”?

Fronting the object for focus, by converting an SVO sentence into an OSV one, is a common enough syntactic pattern in English:

Contest Rules

Submitting ungrammatical sentences is of no use here.
Them we refuse to even consider.

Them?

This is a complete sentence.

At the risk of stating the obvious:

'Them' is the word that starts this sentence.

If that's a little too meta:

'Them' is my favourite movie.

This form (using a gerund, or noun phrase) hasn't been mentioned yet, and is grammatical albeit awkward:

Them being able to come up with such unusual sentences was a surprise to some but not to others.

(Note that "their" — and perhaps "they" too — is also acceptable as the first word, but by no means obligatory, and in fact less common.)

Them was Van Morrison's band in the 60's.

Them Again was the name of their second album.

Them In Reality was the name of their 1971 album.

At least four sentences on the linked wikipedia page meet your question.