You can take the Long Island Railroad to Montauk.
New Jersey Transit will take you to the shore.
There's a bus that will take you to Hunter Mountain to go skiing in the Catskills.
Look up maps of the above 3 train systems. In many cases you can take a train, then bike the last couple miles, and end up in a very very isolated place.
I can also go to places that feel isolated and rural like the Jamaica Bay, the Pelham Bay Park area, or Inwood Hill Park, but are technically not because they're so adjacent to developed areas. But they do feel locally rural if you avoid a view that lets you see the city, because there is a jump discontinuity in density.
Also, you can do traditional car trips and just rent a car for the weekend. City people do this crap all the time. You really think we don't get outdoors, huh?
Demand more from your transit systems!
Personally, I live in the city and like getting out but I just use Gocar (pay by hour car rental)
That being said, owning a car is still a must if you have anything close to an outdoor lifestyle. For example:
Leaving the car at the trailhead several hundred miles from home for the whole weekend.
With Uber and Lyft, I already see my friends living an Urban lifestyle completely dumping their car. When this will become automated, another portion will likely dump their car.
But for now at least, I cannot see the outdoor crowd, or the ones living in a real rural area (think opposite of the SF hipster) dumping their car.