North Quincy is only 7 or 8 minutes longer by T than Savin Hill is to Downtown Boston/Cambridge and in my opinion, it is a very quiet and nice area that is also revitalising as well. Another bonus is that NQHS is said to be a very good high school. I visited Quincy a lot of times since I live in Cambridge, but, the disadvantage is it is not going to be as walkable or bikeable and is more auto dependant than JP, Savin Hill, or Southie. https://www.redfin.com/MA/Boston/70-Grampian-Way-02125/unit-3/home/11504452 Check out this house in Savin Hill. It fits your budget and plus, it is around 1200 sqft.