If you need to be close to New York City for work but you also dream of living in a big house with a backyard and growing your own fruits and vegetables (and maybe raising a few chickens), then you have a few options. While you can certainly find large, authentic farm spreads far north of the city in Sullivan county, that is not convenient for daily commuters. If you are going to travel into Manhattan every morning and head home every evening, you don't want a three hour commute. So, with that said, this list will include two areas close to Manhattan.
West Milford, New Jersey
West Milford is located in Passaic Country, New Jersey. It's located northwest of Manhattan. It's in northern New Jersey and is not very populated. There are many large houses with lots of property. There are even large working farms (onion farms and dairy farms in particular) in the area. The town does have a major shopping center with a large supermarket and other amenities, so you won't be completely "out in the woods" like some areas in northern Jersey. The one downside to the area is that you will either have to drive in to Manhattan or take a bus. NJ Transit does not run train service to the area. Because the area is located in norther New Jersey, you can either drive into Manhattan via the Tappan Zee approach (which will take you south through Westchester and The Bronx) or head in through the George Washington Bridge.
Tottenville, Staten Island
Tottenville is located at the extreme southern end of Staten Island. While it is part of New York City, you would never know it when you walk around the neighborhood. The area is much less densely populated than northern Staten Island and feels like you are in the suburbs. The homes are large, and the space you get will afford you the opportunity to grow a really nice vegetable garden. Plus, it's perfectly fine to raise chickens in the city, and there are those in Staten Island that do so.
There is a train that runs the length of Staten Island and will take you right to the Ferry. The Ferry (the Staten Island Ferry) is free and you can take this right into lower Manhattan. It's steps from Wall Street, so if you work in lower Manhattan (especially the financial district), then it's a perfect commute.
Talk to a real estate agent for more help finding the perfect rural home for you.