Your quotation comes from Charles Dickens's "A Tale of Two Cities".
Age - a distinct period of history;
Era - a long and distinct period of history;
Epoch - a particular period of time in history or a person’s life.
They are similar but there are certain set terms which cannot be changed, such as:
the Stone Age, the Medieval Age, the Age of Enlightenment, the Jazz Age.
We usually say the Victorian age or the Victorian era, but not the Victorian epoch.
Epoch tends to be shorter or on a lesser scale, I think: "that epoch in his life".
Charles Dickens was using the nouns in pairs, and he wanted to do "elegant variation" by rhythmically switching to a new noun after every pair:
"It was the best of times, it was the worst of times, it was the age of wisdom, it was the age of foolishness, it was the epoch of belief, it was the epoch of incredulity, it was the season of Light, it was the season of Darkness, it was the spring of hope, it was the winter of despair…"
So the requirements of art prevailed over precision of meaning. We remember, however, that we are not Dickens, and so we need to be more precise when we write.