I tried to download the "19.8 GB" GOG game "The Wicther 2" (Linux) several times with my broweser (Firefox), but it always breaks after about 5 up to 6 GB download. GOG Galaxy is not available for Linux so far. So what alternative options do I have to download such big files from my GOG library?

Maybe this can help:
https://www.gog.com/forum/general/release_lgogdownloader_wrapper_mkii/page1

Another option would be GoGrepo.
Nice usage of the "quotation mark" BTW.

Alternatively, you could try and find a download manager for Linux that would hook on your browser and download anything through it (or paste the link of the download to it). On Windows I use Free Download Manager which even allows me to change the link in the properties of the download in case it fails entirely and I have to restart. If you were to find a similar software (because FDM isn't available for Linux I'm afraid), it'll be a boon for future downloads.

While I have no experience with the listed applications, there are quite a few listed at alternativeto.net (where I did a search for alternatives to "Free Download Manager" that run on Linux).

Some are better than others, and others might not at all have the feature you're looking for, but going through the list, you might find something that works.

(Yeah. PookaMustard, I replied to yours rather than the OP, sorry about that, I tried to expand on your post.) For some reason I always read that as "Down The Mall", and get the image of someone swallowing an entire mall in one go.

This is a nice firefox download manager if you are using. Really neat thing to use
https://addons.mozilla.org/en-US/firefox/addon/downthemall/

Nah, that's okay. Feel free to quote any and all posts of mine. With that said, there's a reason why I didn't bring up a name of an alternative myself, simply because I have no experience with them enough to tell a story about just how good the thing is.

If anything though, I think Free Download Manager won't mind running with WINE?

As others have said, use a download manager. You could go for DownThemAll or maybe jDownloader or even install a Firefox extension called FlashGot (has nothing to do with Adobe Flash) and just choose whatever manager you wish to use - you probably already have wget installed, for example, and it's all you need.

This is why I suggested FlashGot. You can do it with just wget, but you need to give it the GOG cookie (--load-cookies switch). However, Firefox stores the cookies in a SQLite database, so you'll need to first retrieve it from there, probably using some extension. And if you're installing extensions anyway, well then, just use FlashGot (or DownThemAll).

You need to pass wget a valid GOG cookie in order to download something with it.

Extract the gog-al cookie from your browser.

--- cut here ---

#!/bin/bash

cookie="gog-al=PLACE_YOUR_COOKIE_HERE"

for i in "$@"
do
ofname=$(curl -s -b "$cookie" -I "$i" | grep Location | sed -e 's/Location: \(.*\)?\(.*\)/\1/')
ofname=${ofname##*/}

echo Download: $ofname
curl -L -C - -b "$cookie" "$i" -o "$ofname"
done

--- cut here ---

Needs curl installed.
Usage: ./script.sh https://www.gog.com/downlink/the_witcher_2/en3installer1

You can place several links in a row if you like. It will download them all. Resume is supported by curl itself.