Would it be possible to download old versions of games that are no longer available for purchase?
The DotEmu versions of both KOF98 (Now only available in the Ultimate Match version by Code Mystics) and Garou (same) for example.

KOF98 was never available in a non-Ultimate version here on GOG as far as I know...

As for Garou MotW, the Linux installer still contains the old (emulated) version of the game and the original ROM.

google.com

On a case-by-case basis. Usually, the store page indicates if a game also offers the older version (like Simon the Sorcerer). In addition, provided that the older version was available at one point here and got replaced (like Little Big Adventure), you can access it via the Galaxy client's rollback feature. Unfortunately, I don't think there is a similar option for offline installers.

You would have to use the “optional” client if at all possible. There is no old versions (even if the new one removes content or breaks things), nor is there always patches, or changelogs.
https://www.gog.com/forum/general/provide_a_full_and_complete_changelogged_download_system/page1
It’s unlikely to change as there is zero interest in offline installers or anything offline.

Thanks for the answer! I just downloaded the Linux version of Garou but all I got was a .sh file. Is that what you're talking about or did I do something wrong?

The .sh file is the Linux installer provided by GOG.
You can extract its content using tools like bsdtar or unzip.

If you engage in illegal piracy, then yes.

If you do not engage in illegal piracy, then no.

You'll need to unzip it to get the ROM ;). And of course you won't be able to run the Linux version on Windows, but you can use the ROM in your emulator of choice, such as MAME or fbneo. ... or you can engage in resonably legal piracy :P. If you own the Ultimate Edition Final Cut of KOF'98, I see no problem with getting the ROM for the original from other sources. It would have been nice for it to be included as a bonus in the first place...

Sometimes GOG provide access to an older or original version in the Extras section at right on your library game page. Usually they mention such on the game web page. Failing that, unless you can somehow roll-back using Galaxy to an old enough version, then you are out of luck at GOG.

The only other recourse would be another store or the publisher or developer website, or you need to go the iffy abandonware site route. There are many abandonware sites out there, but the legality of them is dicey to say the least.

At the end of the day with Abandonware, it is down to you and your conscience and what you can convince yourself of. Certainly if you actually own the game, I cannot see why you wouldn't be entitled to get an older version if you want.

Besides the tools vv221 mentioned, if you only have a Windows machine, iirc, 7-Zip should be able to extract the game's contents too.
And if you don't have either a GNU/Linux machine or an OSX machine but still want to use the tool "unzip", it should still be possible to be done if you install Termux on Android and then unzip within it, or Windows Subsystem for Linux on Windows then install unzip in it. And I think I have a Python script lying somewhere that can do unzip's functionalities without having unzip installed, but I need to find it again.