Here is the LI realty website.
Find a home in Nassau, Suffolk & Queens | MLSLI.com - Long Island Real Estate Select a town, like Franklin Square for example, if you go to the detail section and scroll down you will see the lot size. As I and others have already stated, you will find virtually nothing in your price range in Nassau County and while I am not all that familiar with Suffolk County, I doubt that you will have much success there either. \
Long Island is a tough market (read expensive) for those just starting out and with your desire for privacy the "wish list" has now increased the cost exponentially. Cost aside, Long Island is very densely populated, in fact years ago it was more populated than 2/3 of the rest of the US states. The history of LI housing was seeded in Levittown just after WW II (google Levittown) and affordable homes were the goal for the masses.
I would suggest that you look toward much more rural areas like upstate NY for your wish list. Westchester is just as bad as LI. Of course, your spouses' job may dictate a change in either your wish list or pocketbook as you need to live near where your job is or near to where you can find employment.