Verify that the external hard drive hasn’t swapped its drive letter since installing Plex server. This can quite easily happen if you juggle around with external storage like USB hard drives and memory sticks. Of course there is also a method to prevent it: assign a static drive letter.
Plex server closing itself is usually caused by a problem with the database file. If it’s either damaged or completely missing.
Take a look into the Windows registry where Plex is expecting to find its Plex data folder.
If you have moved the plex data from its default location on drive C: to your external hard drive, and that drive has since changed its drive letter, then Plex server closing itself is to be expected.
If there is a discrepancy between the registry key and the actual location of the plex data folder, change the registry entry.
Recommended: change the drive letter of the external drive to a static drive letter first.
If the external drive is a “mobile” hard drive (usually advertised as power-saving), then it should not be used to house the Plex data folder. These drives tend to ignore the Windows power management and spin down their platters whenever they deem it right. Which then leads to longer timeouts and potentially to database damages in Plex.