Had this problem when my storage filled up

systemd will try to save any kernel panics in pstore via systemd-pstore, if it's enabled in your kernel. I'd check /var/lib/systemd/pstore and see if anything in there on the next boot.

Can you elaborate on ping "running"? Do you get actual icmp replies coming back? Because there's no code path I can imagine where a ping would cause hdd activity (on a normally running system).

If ssh times out (and you don’t do anything fancy with the firewall), then it's not sshd dead, it's sshd not being able to respond. Grab a tcpdump for dst port 22 from your local machine while RPi is stuck and see if you get any replies whatsoever or it's just retransmits going into the void.

My first rough suspicion would be ram abuse. Something eats up all ram and the system locks up and semi-dies. Pstore would have OOMs. You could run a local script for telemetry recording too to see if ram use spikes up before the system gets unresponsive.

It sounds to me like ssdh may have stopped working. That may explain why you can't ssh into your server but pings still respond. I have a Raspberry Pi4 and a Pi5 and have had similar issues in the past.

I would probably approach this issue by writing a small script that checks every so often if the process sshd is still alive and if not restart sshd. Maybe SystemD can so something similar but I am not familiar with SystemD.

Edit: A quick and simple script looks like this

#!/bin/sh

# Check if `sshd` process is running; If not running, `pgrep` returns
# an exit status of '1' and restarts `sshd`
pgrep 'sshd' > /dev/null || systemctl restart sshd

Make the script executable with chmod +x /home/user_name/sshd_check

Add the following line to /etc/crontabs/root to run the script every 15 minutes

 */15    *       *       *       *       /home/user_name/sshd-check

I don't use SystemD but I am pretty sure systemctl restart sshd is correct, otherwise it can be changed to whatever your operating system uses to control services