This is an excellent question.
I currently run OpenHAB on my Synology (there is a specific topic on this). If you have a Synology, you know this is a very stable solution. An attention point is that, when you upgrade/patch your Synology to a higher DSM, you often need to reboot, reducing (OpenHAB) uptime. In my case, I use my Synology for many other things as well (e.g. Plex Media Server), so this could negatively impact OpenHAB performance (however, this has not yet happened).
I think stability/reliability does not only come from hardware, but also software. For using OpenHAB at a Synology, I see this as an attention point. There is no clear upgrade-process (no apt-get available) on Synology for OpenHAB, and sometimes you need to be a bit creative to get additional software running. E.g. InfluxDB (for persistence) runs via a Docker image. That’s why I am reconsidering to use my Synology for OpenHAB.
Many people here use Raspberry Pi. The good thing is that in terms of software, many things are standardized. With OpenHABian it is almost “plug and play”. An attention point here is that the SD card could get corrupted after a while, if you have a lot of read/writes (e.g. with persistence services). That is a no-go for me. Alternatively, you could boot OpenHABian via an SSD which is much more reliable. But the firmware needed for that is still beta. I guess that conflicts with the requirement “stable/reliable”.
So in terms of hardware, I am still looking for a good solution. I’ll keep an eye on this topic.
In terms of OpenHAB version, I use a snapshot version of OpenHAB 2.1. I have read in earlier posts that it has the same reliability as OpenHAB 2.0. I did not consider a 1.x version. They might be more stable, but there are some fundamental differences between the 1.x and the 2.x and I didn’t want to have all that rework when upgrading to 2.x at some point in time. So far, my experience with OpenHAB 2.1 has been good in terms of reliability.