Your data is completely safe when you shut down a server. All data is stored on persistent virtual drives that exist independently of the server instance.
When you power off a server, only the CPU and memory resources are released. Your drives, and all the data on them, remain exactly as they were. When you start the server again, it boots from the same drives with all your files intact.
This persistence means you can safely shut down servers to save on processing costs without worrying about data loss. You will only be charged a small holding fee for the storage space your drives occupy while the server is off.