It is not true that neural networks are only suitable for big datasets. It is just that modern deep architectures that are trained from scratch require relatively large datasets.

But there are plenty of alternatives if you want to do ML with small datasets. Use pre-trained networks if possible. Use smaller network architectures. Or don’t use neural networks at all. There are lots of approaches from the era before deep neural networks (e.g. support vector machines, decision trees, probabilistic models, like Bayesian networks) and they are not obsolete.

Choose the right approach for the problem and don’t think that because deep neural networks are the new hotness, that you need a neural network.