I was living on 10th Street off Hudson.
https://collocations.enacademic.com/17693/street
What does "off Hudson" mean?
Hudson Street crosses 10th Street. You were living in the block of 10th Street between Hudson and the next cross street, probably in the half closest to Hudson Street. You might be on either side of Hudson, unless 10th Street ends at Hudson, of course.
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Hudson Street crosses 10th Street. You were living in the block of 10th Street between Hudson and the next cross street, probably in the half closest to Hudson Street. You might be on either side of Hudson, unless 10th Street ends at Hudson, of course. It is a rather imprecise expression.