Owning a private island fosters a certain fantasy. Whether the draw is anonymity, power, or being surrounded by turquoise waters, the perks of getting away from it all can be coupled with challenges. Many islands are quite difficult to reach and can prove impossible to develop.

While the 26-acre Pumpkin Key is remote, it is easy to access. Past owners have installed water and power lines from nearby Key Largo, just 10 minutes east by boat. The three-bedroom, 5,000-square-foot main home; two neighboring care…