Are your backups running during these lockouts?
It's kind of curious cause it sounds like an i/o problem, but if it was, your mac would feel sluggish even when you were logged in locally.
That's why I asked where your home directory was... if your remote ssh is blocked waiting for a device or something on log in (something i've seen happen with user home directories mounted over NFS) it'll just hang there even though the system itself is ok.
yes, i reckon, too. forgot to mention: once the error occurred, the server cannot shutdown or reboot no more, even locally.
generally, the filesys control seems unstable; some 10 days ago an internal data drive went offline..no log entries again apart from the system starting to whine that it lost a drive...
remount failed, disktuil repair-volume function came up with a thoroughly corrupted and unrepairable filesystem... had to reformat.. no idea if that's related, this happened on its own outside the usual afp/ssh problem hours.
i got daily ccc-clones running on that 4tb drive plus tmachine, running currently in its default configuration, ie. hourly. so it's well possible that tmachine had been running when the srv went deaf and dumb...
What do your system load scores look like?
what does SSH's state look like when you do a ps aux | grep ssh?
do you have anything like fail2ban running? maybe it's a firewall thing...
what does ssh say in the system logs?
what does your ssh client say when you run it verbosely?
sys load is usually low, and stays low after it goes dark.
no fail2ban or anything, trying to keep it simple here.
no ssh log entries after it goes dark, like i said, i'm not even getting a password prompt...
never tried -v when trying to login remotely...