A stone's throw away from Manhattan, Bronx, and LaGuardia Airport sits North Brother Island, an one-time quarantine facility that housed infectious individuals like "Typhoid" Mary Mallon. The island has since been exiled from the surrounding metropolis and left to nature.
Despite being in the East River (and basically next door to Rikers Island), North Brother Island has been uninhabited since the Sixties; it's presently used as a bird sanctuary by the New York City Parks Department. The island's buildings have been left to rot for almost 50 years. Kingston Lounge has a detailed history and many photos of this abandoned facility:
Situated in the Hell Gate, a particularly treacherous stretch of the East River, North Brother was home to the quarantine hospital that housed Typhoid Mary, was the final destination of the General Slocum during its tragic final voyage, and was the site of an experimental drug treatment program which failed due to corruption. Riverside Hospital, the name of the facility on the island throughout its various incarnations, treated everything from smallpox and leprosy to venereal disease and heroin addiction; after the Second World War, it housed soldiers who were studying under the GI bill. The entirety of the island has been abandoned since 1963; over a dozen buildings remain, in various states of disrepair.
For another piece of creepy abandoned real estate within New York city limits, check out the South Edgemere Wasteland of Queens.