I purchased a few games during the last sale that when they were released would work on my oldest Windows computer, a Dell Optiplex GX110 that dual-boots Windows 98SE and Windows XP. with 512MB of RAM and a single-core 1ghz processor.
However, GoG uses a customized installer and seems to also have some modified versions to let the games work on Windows 10, often installing DOSBox for really old stuff.

Would I be able to install the GoG version of the original Fallout on my old system so I can play it while I'm doing my annual backup and full virus/malware scans of the newer computers?

I have no intention of buying these from GoG since I already own the CD versions, but if it's a question of hardware, figure the same question for any of the following games. If the answer is "Yes" to the GoG version of any of these, then Fallout should work:
Baldur's Gate I or II (Not EE/Enhanced Edition, the original)
Arcanum: Of Steamworks and Magic Obscura
Icewind Dale I (NOT EE)
Icewind Dale II
Star Trek:Starfleet Command
Heroes of Might and Magic

Note that I have had these games (except Fallout) installed on that computer for at least 10 years, but they are the original CD versions, not the GoG versions. I know the originals work fine, I just don't know if I can install the GoG version on an old XP or 98SE computer even though Fallout 1 is old enough it should work.

Thanks, it says XP/Vista/7/8/10. I'll give it a try.
I was concerned the modified versions wouldn't work on my old hardware and I'm planning on tying up my new computer for a while, thought I could play one of the classics I missed back in the day on period hardware while I did the annual maintenance on the new stuff. .

Although that modification, does that mean that if I bought "SimCity 3000 Unlimited" from here, I might actually be able to play it rather than encounter the same problems I had with the CD version? I've had the CD for quite some time, but I could never get the darn thing to work for more than 5 minutes before it crashed.
The CD version won't work with Windows 10 because of the copy protection, and I couldn't get it to run for more than 5 minutes before crashing on the aforementioned 98SE/XP Pro machine or the Vista Home Premium/7 Professional computer I have (I used to need to VPN to a domain).
I don't particularly care which computer it runs on.
On the other hand, I have SimCity 4 (gift from somebody), and a friend gave me his copy of the Mac versions of SimCity 1 & SimCity 2000 when he switched to Linux and Windows that I'm sure I have somewhere in a box in mom's basement if I still had a working Mac (my 6500/300 died in 2008 and I decided it wasn't worth spending $90 for a new proprietary power supply when I only used it for the TV tuner card or to display stuff from GameFAQs when I got stuck gaming), so maybe I won't bother wasting the money.

The original point of GOG was old games made to work on new hardware with no DRM, so yes SimCity 3000 should work on Windows 10 without issue. Though with PC gaming you can never really guarantee that in 100% of cases, as I'm sure you know.

I know, I've got two games that I could not get to work on ANY computer I own. Fortunately I did not pay a lot of money for them. I found them both at the library "book sale": SimCity 3000 Unlimited and Star Trek: Dominion Wars.
Both have much the same problem. They appear to install just fine on the Optiplex GX110 and start up. Both crash within 5 minutes of launching.
Both refuse to start at all under Vista or Windows 7. I don't think I tried either on the Windows 10 computer as I only got that last November. I read about the problems with SC3K on Windows 8/10.
I've researched the problem and tried every trick and launcher out there. Nothing works. One of the patches was needed to get SC3K to launch under XP.
Well, thanks you for the advice. Is there a way to mark this as "Solved"? I don't see a way to edit the subject.

There should be a checkbox with something like "choose this post as the solution to your problem" next to it.
That does the trick.