Not all stateful services should be treated in the same way. Some might need an external drive mounted, while others already have some kind of a replication and synchronization incorporated. In some cases, you might want to combine both mounting and replication, while in others replication itself is enough.
Please keep in mind that there are many other combinations we did not explore.
The important thing is to understand how a service works and how it was designed to persist its state. In many cases, the logic of the solution is the same no matter whether we use containers or not. Containers often do not make things different, only easier.
With the right approach, there is no reason why stateful services would not be cloud-friendly, fault tolerant, with high availability, scalable, and so on. The major question is whether you want to manage them yourself or you'd prefer leaving it to your cloud computing provider (if you use one). The important thing is that you got a glimpse how...