Translation:The men do not know if their wives are eating.
I don't think I agree with you, goedjn, that a notional "the men don't know, whether the wives are eating" would have the meaning you suggest -- chiefly because a sentence so punctuated would not even be English!
Regarding the comma in the Danish sentence, that needn't be bothersome so long as you take into account the "rule" that in Danish commas are inserted between grammatical clauses. Mændene ved ikke is clause 1; deres koner spiser is clause 2 (and om (=whether) conjoins the two).
Now, the only qualification to what I've just written is that if the subordinate (a.k.a. dependent) clause (the one headed by at, da, fordi, mens, når, om, siden, som, så etc. or a "hv-word" like hvad or hvor) comes second then the comma is OPTIONAL. This "revolutionary" :) deviation from previous practice was introduced by Dansk Sprognævn – the Danish Language Council – in, I believe, 2004. It allows you to write EITHER Mændene ved ikke, om deres koner spiser OR Mændene ved ikke om deres koner spiser.
P.S. Should the dependent clause, however, come first -- Om deres koner spiser, ved mændene ikke -- then a comma MUST be inserted. This is a bit of a "belt-and-braces" approach, to tell the truth, since the inversion of mændene ved to ved mændene in the main clause is already an indication in itself that the whole of Om deres koner spiser is a subordinate/dependent clause.
Helpful? Or "clear as mud"? In either case, any reports of faulty analysis will be gratefully received.