Here is a view of the console terminal attached to this Sun SPARC IPX, which was up since 24 August 1998:
As you can see from the ps output there wasn't a lot going on, which is a big reason this computer was able to stay up for so long. A lack of power outages helps, too. :-) The computers we actually rely upon for ARPA.NET services run newer operating systems on newer hardware, and they are rebooted as needed to complete the installation of kernel patches.
Regrettably, just 9 days after this event an unplanned power outage in the machine room put an end to the uptime experiment. NS0.ARPA.NET has since been retired.
Want so see more? Download an 11-second MPEG movie (1.5 MB) showing the login and uptime command in the above picture.