I can see how these 20 or so kids might be the only ones left. Disease, fighting, accidents, starvation, shock may have killed off everyone else in 300 years.
From the episode I got the impression that the children were in a medium sized city at least. If they were eating canned goods from stores, restaurants and private homes that would hold them for decades. Occasional moving on to the next city or town.
And there did not seem to be very many of them, so maybe the illness killed off a lot of the children too, except those with some kind of immunity that protected them through to puberty.