I was stuck with Windows 98 SE for my primary system long past the end of Microsoft's support for it. I can tell you right now, security is actually the LEAST concern when Microsoft ends support for an OS.
The biggest concern is both driver support and software support.
The first problem I ran into following the end of Windows 98 SE's life-cycle is that newer games stopped working altogether, despite my system meeting the requirements, even if just barely. Once I found out about KernelEx I was able to circumvent SOME of these limitations... but that was just the minor issue.
The second problem I ran into was a little more insidious: Things like web browsers, plugins for browsers, firewall software and virus scanners, all of them eventually reached points where the producers called it quits and gave up on Windows 98 support because the developers could no longer get help from Microsoft when they ran into issues, and because they were essentially spending time coding for a platform only a tiny fraction of users were still using, so it didn't make sense from a business perspective. For browsers and for firewall software, you could just stick with the older stuff and be OK... but virus scanners would refuse to update their virus definitions and browser plugins not only started to malfunction with stuff made for later versions of those plugins, some of them were outright crashing the entire system when newer files tried to run inside those plugins, forcing a restart, even when using browsers that were still being kept up to date. (Opera was one of the last browsers to end Win98 support.) Every time I watched a video online there was a 20% chance I'd trigger an instant reboot of the computer. :P
The third problem took awhile to run into and that was driver updates. In order to run newer stuff, your drivers need to be in good shape, otherwise you run into all sorts of tiny problems which degrade the quality of the software you're trying to run, such as by causing blurry/missing textures in games or sound drop-outs in anything audio-related, or in worst case scenarios, programs simply just don't do what they're supposed to.
It took about three years following the end of Win98 support for it to get really bad, though problems started cropping up as early as one year following. Fortunately, I got my WinXP system from my father in either late 2009 (or early 2010, forget exactly when) and I barely touched my Win98 system following.
TL;DR: Appel isn't crazy, just misguided. For a closed network with no internet access and no software updates, he'd be fine. However, after a year or so, accessing the internet and attempting to run newer programs WILL screw him over if he plans to do those things. :P
--- Kris Asick (Gemini)
--- Pixelmusement Website: www.pixelships.com
--- Ancient DOS Games Webshow: www.pixelships.com/adg