It is true that neural networks are inherently not good at classifying 'unknowns' because they tend to overfit to the data that they have been trained on, if the underlying structure of the neural network is complex enough. However, there are multiple ways to go about reducing the affects of overfitting. For example, one technique that is used for this is called dropout. Another example can be batch normalization. Despite these techniques, the best way to reduce the affects of overfitting is to use more data.
For the facial recognition example that you have given above, it is common that the models that have been trained have 'seen' a huge amount of data. This means that there are very few 'unknowns' and even if there are, the neural network has learned how to tell if there are facial features present or not. This is because certain structures of neural networks are really good at telling if there is a pattern of features present in the input data. This helps the neural networks to learn if the image that is being input has certain features/patterns in it or not. If the these features are found then the input data is classified as face otherwise it is not.
What can you do in your case?
Let us assume that you are going to train your neural network to recognize if an input image is a cat or not. You will use a Convolutional Neural Network (CNN) and train it to recognize if the input is cat or not. The not part means that you have to include a lot of examples in your training data that are not cat. In the perfect case you will be able to show it everything that is not a cat and classify it as such. Also you show it multiple images of what a cat is. CNNs are really great for this application. You might want to research regarding this and see what kind of CNN best suits your application. If you don't have gazillion samples of what a cat is not then you can use regularization techniques like dropout and batch normalization.
PS: For more details please mention what strategies you have used up till now. Also it would be better if you can share what your desired task is.