Is your NAS running DSM 6 or DSM 7?
Edit: See Synology FAQ 15: Claiming & Connecting to a troublesome Synology server.
It has a procedure for both DSM 6 and DSM 7.
Claiming a server requires communications with hosts at plex.tv.
Those hosts are slammed right now.
So, no guarantees the claim process will work until the load on those systems subsides.
The process in the FAQ does not touch your media, server settings, etc.
If it does not work, wait awhile and try again later.