The Amazons were not a race
or a nation
, because they did not reproduce. New Amazons were not born, they were dedicated.
As far as the sketchy historical record will let us penetrate the period of time in which the Amazons were most active (pre-Homeric Asia Minor, so we're talking western Turkey in the first half of the first millennium BC), they were priestesses of Artemis/Ishtar, dedicated to her service and virginal (hence the man-hating stigma).
They had their own Polis, Thermagont (I'm not very sure about this transliteration, so apologies in advance), in which there were in fact quite a few men - mostly blacksmiths, grooms, dog trainers and other labourers. Many of them were quite high-born, and their dedication usually entailed a generous gift to the temple of Artemis by their families, so the Amazons rarely lacked finance, and were little inclined to be the marauding viragos they are sometimes made out to be.
The myth about the Amazons cutting off one breast comes from the way that the Greeks of the period used their bows - much like people do today, they shot with the bow to one side of the body. With such a position the breast on the side of the dominant arm would indeed be a hindrance (I sure as hell wouldn't want a rebounding bowstring to hit one of mine!). However, the Amazons used smaller, more steeply curved bows than the Greeks, and shot stright ahead of themselves so that the string would rebound towards the center of their chests.
In Classical Helenic culture, and later during the Renaissance and Neo-Classical revivals, the Amazons, who were great horsewomen and cavalry soldiers, are depicted off their horses, armour lying in the dust, being dragged by the hair or otherwise subjugated by Greek mythical heroes (Jason, Theseus, Heracles - they all supposedly had a stint at subjugating the Amazons). From the titles of the works you can see that the men are often in the process of avenging a previous humuliating defeat - but of said defeats the artefactual record presents not a single scene.
As a potentially interesting aside, I have recently taken up the longbow, and I find that provided one follows instructions the risk of hurting one's breast is quite low. In my neophitic bumblings I managed to catch a nipple on the backtwang once only, and even that didn't hurt as badly as you'd think. To me, this makes the mystery regarding the Amazons' breasts even greater. Then again, Greek bows may have been sufficiently different to Welsh longbows to have given rise to this rather grisly misconception.