He was only twenty-two then but had already been in two or three of what were spoken of in Winesburg as "women scrapes."
Scrape is used here as a noun, where it means a difficult or embarrassing situation, usually caused by your own actions.
A 'woman scrape' is therefore a difficult situation involving a woman. Further down the text it refers to the character 'getting women into trouble' (a euphemism for getting them pregnant). So these 'scrapes' are likely to be either getting a women pregnant or being at high risk of doing so.