I was also unable to send email notifications from my Linkstation 420D after setting up the system for the first time.
After varying the email settings many many times without success, I realized that the problem is the one, which turtle1980 already mentioned: "the NAS drive being unable to access a DNS server...".
Indeed by default my Linkstation Default Gateway address and Primary (and Secondary) DNS server addresses were blank.
So for you others, who still have problems, check and set them, if the fields are empty.
Open the web setting access of your Linkstation and navigate to :
Advanced Settings -> Network -> IP Address
There click the "change" button to change:
Default Gateway address
Primary DNS Server
(Secondary DNS Server)
I guess you can put there your Internet service provider's DSNS and Gateway addresses or some general DNS server addresses, but for me everything worked fine, when I have set them to be my own router's address in my own network:
Default Gateway address 192.168.1.1
Primary DNS Server 192.168.1.1
I even left the seconday DNS Server address blank, because if my own gateway is broken, there is no other way out.
I hope this helps.
Btw. This solved also my problem of NTP-server updates, which were also not working, when the email notification was "broken".