This quote refers to Oliver's poor mother and the way in which she met her death whilst giving birth to her son, the aforementioned Oliver Twist. The quote you have identified concerns the somewhat ignoble situation of Oliver's mother. She, suffering from reduced circumstances, ended up in a work house, which was a kind of Victorian charity house for beggars who could not look after themselves. She gives birth in this workhouse and as such receives minimal medical care from the surgeon. However, because of the nature of the workhouse, many women go there who have become pregnant outside of marriage. This is what is referred to as "the old story" by the surgeon as he tries to see what identifying marks there may be on the corpse of Oliver's mother. The fact is that to have a child outside of wedlock was such a social stigma that many women fled their homes and families to give birth in secret. The surgeon assumes that this is another such case.