One difference between Blender and a typical game engine, is that game engines only render one side of a polygon, not both. You need to make sure that your polygons are all facing the right direction. What I think you are seeing here is all of the faces in your model pointing towards the inside, whereas you are expecting them to point toward the outside.
Look at your face normals in Blender- the side of the face that has the normal is the one that you will see in Unity. Blender includes several features for correcting the direction of the normals if you find that they are wrong.