There wasn't a fence around the Amelia Island lighthouse when I was there (it was Feb 2005).
Also we went into the Naval Base in Mayport, but I got a big drop of grease on my camera lens and didn't see it until afterwards so those pictures are not too wonderful. The blurry area (because I turned the camera sideways) is on the lower left, and I didn't get the camera straight on the second one.
We could see the outline of the oil house quite clearly, but it doesn't show well in this picture. (When the base was built, the land was raised about 20 feet and filled to make stable runways. The base of the lighthouse including the door is buried underground. The outline of the oil house roof can still be seen on the side of the lighthouse just to the left of the tree above the fence. Access is through a window.)
This one shows the blurry spot really well
A newer lighthouse was built about 2 miles southeast of this one. I didn't realize that it was also on the base, and I didn't get a picture of it