The Unmentionable Times in Ayn Rand's Anthem refer to the times that occurred before the Great Rebirth. Readers can realize that the Unmentionable Times refer to our modern world, particularly the twentieth and twenty-first centuries. This becomes apparent in chapter one when Equality 7-2521 describes the Unmentionable Times:
"They whisper many strange things, of the towers which rose to the sky, in those Unmentionable Times, and of the wagons which moved without horses, and of the lights which burned without flame."
Of course, readers can recognize that Equality 7-2521 is describing skyscrapers, cars, and electricity - all inventions of our modern times. These items, and the time period in which they exist, are unmentionable because Equality 7-2521's society is one in which "all men are one and that there is no will save the will of all men together." Everyone is supposed to be exactly equal in this society, and we all know that being able to afford to live in a skyscraper or drive a fancy car creates inequality among people today. In Anthem, this disparity is why the Great Rebirth occurred and destroyed our current society to backtrack to one that is similar to the Dark Ages, where the playing field was simply bad for everyone instead of just some.
The Unmentionable Times was a time period before that of the Great Rebirth.
During the Unmentionable Times, it is implied that electricity, technology, and individuality existed, and people were allowed to be more superior than others.
This seems very similar to the time period of our history from since society formed to this day.