Would I live in Southbridge as a new grad who is renting? Sure.

Would I buy a house and put down roots? Absolutely fucking not especially if you plan to have kids in public school there.

Look in Sturbridge. We lived there. It's nice, a bit of a tourist spot in summer, but still a bargain compared to the town just northeast of Worcester that we moved to from Sturbridge.

ETA: You won't find a ton under 350, but some do come up.

https://www.zillow.com/homedetails/598-Main-St-Sturbridge-MA-01518/80873487_zpid/