Ax3ra: Could someone help guide me on this problem? I contacted support but it was no help.

Braggadar: Did you download the web installer again like it asked and run it?

Braggadar: How about simply fully uninstalling Galaxy first and then reinstalling?

Braggadar: So you downloaded the web installer, and it failed to install Galaxy properly at all. And you re-downloaded the web installer and tried again?

Hmm.

I'd say it's an issue with the web installer contacting the GOG server. Did it get anywhere through the installation before failing?

If it can't contact the GOG servers itself the installer is going to fail. Potentially this would be due to your firewall blocking the installer proper access, or it's quarantining/deleting files before they can be used. Check your antiviral logs and look for any indications it has done this.

Braggadar: How stable is your internet connection? Do you get regular drop-outs?

Other than that I wouldn't have a clue, mate.

Just in case you're not aware:
You actually don't need Galaxy for the majority of games from GOG.com, especially if you're playing single-player.
1. Go to https://www.gog.com/account
2. Click on the game in your library
3. Click on Download Offline Backup Game Installers
4. Download each of the installer packages you need
5. Install the game. Play the game after installation is finished.

The client allows for achievements and other things, but it's not necessary if you're just wanting to play the game.

Ax3ra: I have no games in my library because I was not able to link any of my accounts due to the launcher not working.
And I have very stable internet.

Darklocq: I worked out how to get around this problem. It IS firewall stuff. I don't know why GOG's installer isn't smart enough to ask you if you want it to set up Windows Firewall exceptions for the parts it needs (and I would guess that most users don't have this problem because they have local or group policy settings that are lax). I also don't know why GOG Galaxy is an installer inside an installer inside an installer. That's stupid, and looks like a trojan (inside another trojan), which may be why the system is thwarting it.

1. Download the base installer, "GOG_Galaxy_2.0.exe".
2. Run it. It will die with an very unhelpful error message that "something went wrong". Do NOT close the error window yet.
3. Open the Task Manager (Ctrl-Shift-Esc). Find "GOG Galaxy - Web Installer" in the process list, right click it, and pick "Open file location".
4. In Windows Explorer, you will now be in a directory with a name that changes every time; it will be something like: "%USERNAME%\AppData\Local\Temp\GalaxyInstaller_Xxxxx", where "Xxxxx" is random gibberish.
5. Go up one directory level, and copy (not cut) this entire "GalaxyInstaller_Xxxxx", and paste it into your Downloads folder or onto your Desktop (or where ever).
6. Close the "something went wrong" error window. That also closes GOG_Galaxy_2.0.exe, and deletes the original "...\Temp\GalaxyInstaller_Xxxxx" temporary folder.
7. Go into your copied "GalaxyInstaller_Xxxxx", and run "GalaxyInstaller.exe"
8. VERY QUICKLY, copy-paste the GalaxySetup.exe that appears. You must do this before another "Something went wrong" error pops up. You'll know it worked right if you can successfully run that copied application. If you get an error about it not being runnable, you didn't do the paste fast enough. If you can't do it successfully after several tries, start up a bunch of heavy-duty applications to bog your system down to a crawl (like run several major games at once, open 5 browsers, run all Office apps at once, or whatever). That should give you time to copy the file. Then you can close all those apps down.
9. Dismiss any open "Something went wrong" error.
10. Run your "GalaxySetup - Copy.exe", probably as an administrator, since this is the real installer. This will, finally, install "C:\Program Files (x86)\GOG Galaxy\".
11a. open Windows Firewall ("Windows Defender Firewall with Advanced Security" – name may vary by OS version). Click "Outbound Rules" (you may have to first pick "Advanced settings" to get to this part, if you are coming in through "Control Panel\All Control Panel Items\Windows Defender Firewall"), then "New Rule...". Select Program, click Next, select "This program path:", navigate to "C:\Program Files (x86)\GOG Galaxy\GalaxyClient.exe", press Open, click Next, select "Allow the connection", click Next, turn on at least the first two of these, or all three if this is on a laptop, click Next, give the rule a sensible name like "GOG GalaxyClient.exe". Repeat this process for "C:\Program Files (x86)\GOG Galaxy\GalaxyClient Helper.exe", and "GalaxyClientService.exe" in the same folder. It does not appear necessary (or desirable) to make matching Inbound Rules entries (GOG has no legit reason to be reaching out and forcing stuff onto your system). Close Windows Firewall when done.
11b. Alternatively, you can open "Allow an app through Windows Firewall" (Control Panel\System and Security\Windows Defender Firewall\Allowed apps), click "Change settings", click "Allow another app...", click "Browse", navigate to "C:\Program Files (x86)\GOG Galaxy\GalaxyClient.exe", click Open, click Add, find "GOG Galaxy" in the list, and make sure it is enabled and permits at least Private (also Public, if on a laptop). Repeat this for "GalaxyClient Helper.exe" (shows up in the list as "GOG Galaxy Installer"), and "GalaxyClientService.exe". Click OK to close "Allowed apps".
11c. Alternatively, if you use some third-party firewall app, do the roughly equivalent steps to enable these GOG applications to communicate with websites. Without doing packet sniffing, we don't know what exact domain names or IP addresses they want to communicate with, and given that the company keeps changing them, that is not likely to ever be a finite list anyway. Just let the apps do what they want when it comes to going online.
12. In "C:\ProgramData\Microsoft\Windows\Start Menu\Programs\GOG.com\", right-click on the "GOG Galaxy" shortcut, pick Properties, in the Shortcut tab click Advanced, and turn on "Run as administrator"; click OK, then OK. This step may not be strictly necessary, but as Galaxy is going to give you options about where to install games, and some locations may require admin access, it is best if this game management app, like Steam, runs as administrator. (If you can't see "C:\ProgramData\" at all, you need to go to "File Explorer Options" (via Control Panel or Start menu search), pick the View tab, then turn on "Show hidden files, folders, and drives", and turn off "Hide protected operating system files".)
13. Now run your installed GOG Galaxy from Start > Programs > GOG.com, log in, and install your games and stuff. It should also work with any download net installers for specific games ("GOG_Galaxy_The_Witcher_Enhanced_Edition.exe", etc.). Try one to make sure.
14. You can now throw away "GOG_Galaxy_2.0.exe" and your copy of "GalaxyInstaller_Xxxxx".
15. After you install a game, you might have to also add a firewall rule for it specifically, though hopefully the GOG Galaxy installers are smart enough to do this for you. If they're not (in general or for your particular firewall), be aware that some games have more than one executable.

[Revised 2020-01-03.]

Hope this helps.