Can you get around Chelsea using public transportation?

Not really. The Dutchess County Loop Bus does run through the general area on Route 9D and there is a stop nearby, but that's about as close as Chelsea gets to having public transportation. Train lines (Amtrak & Metro North) run through here as well, but there is no stop in Chelsea -- you'll either have to go to New Hamburg to the north or Beacon to the south.

That's just for getting around the area with Chelsea as a point of origin; just about anyone living in Chelsea needs a car to get anywhere.