In brief: In the world of web hosting, uptime is everything. Although this example won't set any records for how long it has been running since its last reboot, it is impressive nevertheless given its age and upgrades.
Software engineer Michael Brutman's personal website is hosted on an IBM PCjr (pronounced PC junior), a machine that launched way back in March of 1984. Brutman's 39-year-old home computer is powered by a NEC V20 CPU clocked at 4.77 MHz and has been modified to accommodate 736 KB of memory (IBM only designed the system with 128 KB in mind).
Other noteworthy mods include a sidecar board that allows the PCjr to boot from an IDE drive and a SATA/IDE bridge, allowing him to run a 240 GB SATA SSD on the ancient computer. The modded box also runs a Xircom PE3-10BT parallel-port Ethernet adapter.
According to the server status page, it went online on March 31 and has been running non-stop ever since. That works out to around 2,590 hours as of this writing, or about 108 days. Again, that's far from groundbreaking but is still pretty neat for a system that is almost four decades old.
According to Guinness World Records, the record for the longest period of continual operation for a computer belongs to the Computer Command System (CCS) onboard NASA's Voyager 2 spacecraft. Voyager 2 left Earth on August 20, 1977, just 16 days before its twin, Voyager 1. That predates the existence of the PCjr by several years.
BrutmanLabs.org is a hobby site that, until now, likely hasn't seen a whole lot of traffic. On the site, you will find details about Brutman's various tech interests including the extensive mods to his PCjr. The site is loading pretty slow right now, no doubt due to the current influx of traffic from stories like this.
What's the longest you've had a computer up and running continuously without rebooting? I shut down my system daily so perhaps a full weekend is my record.