Is it possible to move to NYC with just $800?
You can probably do the actual moving for that. But you can't live on $800 in NYC for any meaningful length of time. If you want to live in NYC, particularly Manhattan, you need a good job lined up ahead of time and money saved up - more than $800. Many of the service workers in NYC actually live well outside the city and commute into it for work. You need a good and steady income to live there.
BWAHAHAHAHAHAHA!!! Seriously? $800?!?!?!?! NO
I certainly would not try that, since you will need a minimum of 2 months rent for any apartment (first month + one month deposit), plus food, services and transportation until income starts coming in.
Its possible. It's just not smart.