Your dvr is very out of date. Click Check for Update, then click the Log tab and see if any errors appear
States up to date but logs show an error trying to connect to the update server. Strange. Guessing this is affecting authentication as well - going to reset/re-provision my ONT since my network/internet connectivity seems normal otherwise.
(and yes, my synology is old, you have been more than accommodating making this work for me - thank you!)
-Dean
What is the error message shown regarding update
2000/12/31 17:15:49 [SYS] Error checking for update: Get https://channels-dvr.s3.amazonaws.com/latest.json: dial tcp: lookup channels-dvr.s3.amazonaws.com on 192.168.1.1:53: read udp 192.168.254.19:32907->192.168.1.1:53: i/o timeout
That means your dns is not setup correctly preventing internet access
What I thought as well... but long story short, and resetting nas network settings twice, we traced an error in synology's control panel: regional options: time: time setting: synchronize with ntp server, was unable to connect to the standard time.nist.gov server, and therefore would not allow any connections to be made outside the LAN.
NONE of this was reported in any of the synology's log files 
.
Changed the NTP to time.google.com and the nas (and channelsdvr) is back in operation.
Sorry for the bother!
-Dean