If anyone knows of....

Do a streeteasy search.
Type in 2 bedroom/ location- all downtown/ min price 500K max price 1.3K
click on the box "amenities" and check the box "private outdoor space"

sample search

Sales in Chelsea, East Village, Flatiron, Gramercy Park, Greenwich Village, Little Italy, Lower East Side, Noho, Nolita, Soho, Stuyvestant Town/PCV, West Village, Midtown, Midtown South or Midtown West

We found 84 listings for between $500,000.00 and $1,250,000.00 with at least 2 bedrooms with private outdoor space

coogie, add "terrace" in "description includes". There is nothing under $1M, though.

Well if you add terrace, you'll sometimes miss an add if they forgot to check it off.
Private outdoor space is wider range and worthwhile to go through the ads so you don't miss one.
The field for outdoor space greatly diminshes available inventory to you and you may see a ground floor with a big backyard you like.
Also in this market, expand your search to 1.2 mil as everything is negotiable.

Here a 3 quickies

chelsea
http://www.streeteasy.com/nyc/sale/354494-coop-356-west-23rd-street-chelsea-new-york

murray hill
http://www.streeteasy.com/nyc/sale/371157-coop-333-east-43rd-street-murray-hill-new-york

murray hill
http://www.streeteasy.com/nyc/sale/388909-coop-200-east-36th-street-murray-hill-new-york

UWS under $1M

http://www.streeteasy.com/nyc/sale/441017-coop-33-west-93rd-street-upper-west-side-new-york?email=true

spinnaker1, this is not a real 2 bd (if that's what OP really wants). Also, I happen to know this building. It's a sort of duplex where the stairs —not shown!— take up really a lot of space. And you have to climb to get to the bathroom.

spinnaker1, 33 w 93 is not a real 2bd, you have to climb up to get to the bathroom, the terrace is a piece of the roof and the stairs in the apt. take a tremendous amount of space. I happen to know this building.

Should have looked at the floor plan a little closer. I totally agree, this is a 1/1 with office space -best case.

Real terraces are hard to come by in Manhattan, in my experience. I haven't seen a real deal under $1.1M. Please post if/when you find one.