It could be a failing drive. If you have a spare drive you could reinstall on there and import your guests. Could save hours of troubleshooting software ghosts.

If its only the main OS not responding to SSH and the web portal just disappears after a couple of minutes use, then it might be your router is reusing the IP address linked to Proxmox. Proxmox uses a static IP address, so will not change once set up. Your router on the other hand may re use the address if it was never reserved specifically for the Proxmox machine. In other words, two devices are fighting for the same IP address on your router.

This is what happened to me. Took a while to figure it out until someone suggested checking the IPs and Mac addresses on the router. No issue since fixing it on the router. Just reserve the address on the router, reset the device that is using the same address (if you can figure it out), and reset the router. Make sure you are reserving the address for the correct machine by chcking the Mac address.

Very niche case, but I saw pretty much the same thing.

I was running from an SD card on the iDSDM in a Dell server. The module was flaky and I'd have to power it all off and pop the SD card out and back in then bring it back up.

Unlikely to be the same thing, but maybe some breadcrumbs to start your search

Physically log into it and check the "vitals" (logs and resource usage)

If I had to speculate I would say that your network card has faulty firmware.