Google translate gives "time" as the first translation for "época". Alternatives include era, epoch, period, age. I wouldn't have any complaints about a translation like "He is a man of our season" or "he is a man of our age." The latter, however, may lead to a little confusion. It could be understood as "he is a man the same age as us" which is definitely not the sense in the original Spanish. The translation offered by Duo is absolutely correct.
I don't think Duolingo has anything against cognates (there are hundreds of them here), but I've never heard the expression "he's a man of our epoch." The common expression is English is "a man of our time" or "a man for all seasons. (there was a famous movie named 'A Man for all Seasons'." I'm not sure that expression is often used any more; maybe some of Duo's authors are from my age, era, epoch, time, period, season.
when i looked up epoca in the dictionary it says "period or season" not time. I thought time is tiempo
Would "He is a modern man" work as a translation as well? Would it take away from the meaning of the phrase?
I wrote "Él es un hombre de nuestra época." Could I have also written "Él es un hombre de nuestro época?" What decides what gender the "nuestra/o" is?