Achilles' Bay is located in the northern most part of St. George's(and the whole island). It is just to the southwest of Ft. St. Catherine.
Between Achilles' Bay and Tobacco Bay(which is just a little further south west) is one of the best places to snorkel near short. (The best place being 10 miles north on the reef)
Located at Achilles' Bay is the semi-private beach associated with the St. George's Club. The beach itself is not very large, perhaps 100 some odd yards long. In general though, beaches in Bermuda are not huge. The reason being that the barrier reef blocks the large waves, so not as much sand was formed over the years as would have otherwise.
Also there is Blackbeards. This is the current incarnation of the St. George's Club beach house. It seems to change every couple of years. Sometimes for financial reasons, some time because hurricane's blow it off it's perch on top of the hill.
You can also rent water sports gear there. Nothing big, snorkels and raft's I think are basically it.
This year, we went on the last day(Friday). We were there for about a half hour when it started to rain. Cold rain on the beach is NOT fun. My brother and father went snorkeling as well, but apparently the tide was out, and the water was too shallow in the good snorkeling area.