often connecting to certain home automation technologies requires access to hardware physically located in the home.
Local to LAN for any technology that is locally controllable. Technologies that goes through some vendor’s cloud services can be connected to from anywhere.
Given this, almost no one tries to run openHAB on some sort of external cloud service. Doing so will exclude the option to integrate with whole swaths of home automation technologies.
You could host your own instance of the openHAB Cloud Server which will allow remote access to your locally installed openHAB (assuming you don’t want to use the openHAB Foundation hosted instance at myopenhab.org). But that’s just a sort of reverse proxy to access an openHAB instance hosted in a LAN from the Internet.