Then, read the Layers page in the Unity docs, that explains how to select different layers.
In your case, you would want to do something like this, if your object was on User layer 8:
Camera.main.cullingMask = ~(1 << 8);
Because 8 is the layer mask your "selective" object is on, and the ~ operator reverses all bits, specifying "all but this layer".
And to reset it to render everything:
Camera.main.cullingMask = ~(0);
Edit:
Since you finally described what you wanted, after an incredibly large amount of comments, this will work if you stick it in Update somewhere. Fill in however many layers you want.
This wouldn't so much HIDE stuff on a custom layer but doing something like the following can set it to cull at certain distances taken from: Camera.layerCullDistance page. If, like me, you were just looking to hide things after a certain distance this is much easier. I had been juggling layers in the same way, this is a much cleaner approach to what I was looking for. Hope it helps, if not there are plenty of good suggestions up top.
function Start ()
{
var distances = new float[32];
// Set up layer 10 to cull at 15 meters distance.
// All other layers use the far clip plane distance.
distances[10] = 15;
camera.layerCullDistances = distances;
}