That's what you do. If you're on GCP or AWS, you typically mount a network volume (Persistent Disk on GCP, EBS on AWS).
Kubernetes ensures that the container always has this volume mounted, and of course only one container at a time can claim the volume for itself.
What you should avoid doing is to use a host mount and pin a pod to run on a specific node, because then that pod can only run on that node, and you have no way of migrating without manually moving the mount and unpinning the pod. With Kubernetes, you really want to avoid thinking about nodes at all. State follows pods around; pods don't follow state around.
Kubernetes ensures that the container always has this volume mounted, and of course only one container at a time can claim the volume for itself.
What you should avoid doing is to use a host mount and pin a pod to run on a specific node, because then that pod can only run on that node, and you have no way of migrating without manually moving the mount and unpinning the pod. With Kubernetes, you really want to avoid thinking about nodes at all. State follows pods around; pods don't follow state around.