TL;DR: if you cannot access DSM via the web or the apps, the date on your Synology might be wrong causing the services to either not start or not respond correctly.
I had a strange error with my Synology DS416play the other day. I had to power the system down to connect it to a new UPS. Upon powering it back up, I could not access the DSM console via the web, nor would any of the Synology / DS apps connect. No amount of restarts or kicking the box would fix the issue.
What was strange is SMB and SSH were working fine, meaning that the underlying OS and system were working fine. I could copy files to and from using SMB/CIFS and I could ssh into the box itself.
I opened a support case with Synology and we started collecting logs and other information. One of the things they asked for was to confirm that the drives in the system were on the compatibility list. To do that, I had to shut the system down. When I powered it back up, I noticed that the drive lights had shut off. I confirmed that the system was running and that made me realize that the date could be wrong. I have the lights set to go dark at night.
I logged into the server and checked the date and sure enough, it had reset to Jan 1, 2015 00:00:00. I sudo'd to root and reset the date using the date command and suddenly DSM access was restored. I assume that means the CMOS battery needs to be replaced, but that's something for tomorrow ...