How can I get the full path to an object in an ObjectField?

Path relative to project folder can be found using the following:

 Object selectedObject = EditorGUILayout.ObjectField(selectedObject, typeof(Object));
 if (selectedObject != null) {
     string assetPath = AssetDatabase.GetAssetPath(selectedObject);
 }

Which will produce a string like:

 "Assets/Some/Path/YourObject.prefab" (if you select a prefab)

You can then get the absolute path (if needed) using:

 using System.IO;
 ...
 string filePath = Path.Combine(Directory.GetCurrentDirectory(), assetPath);
 // i.e. "/Your Unity Project/Assets/Some/Path/YourObject.prefab"

 // The following is needed if you are using Windows :-)
 filePath = filePath.Replace("/", "\\");
 // i.e. "C:\Your Unity Project\Assets\Some\Path\YourObject.prefab"

You can use :

 string path = AssetDatabase.GetAssetPath (myObject);

Note: path will be the full path to the object that has been dropped onto "myObject" objectfield.