How many observations in a neural networks dataset?
A neural network is nothing but a set of equations. And the basic rule of any set of equations is that you must have as many data points as the number of parameters.

The parameters of any neural network are its weights and biases.

So that means that as the neural network gets deeper and wider, the number of parameters increase a lot, and so must the data points.

This being said, the more proper and detailed way to know whether the model is overfitting is to check if the validation error is close to the training error. If yes, then the model is working fine. If no, then the model is most likely overfitting and that means that you need to reduce the size of your model or introduce regularization techniques.