This is actually a CDN (i.e. Akamai edge) error. Deep links (i.e. the forums) are working, but the main page and game pages (possibly due to large images?) are not loading for me. Screenshot attached for www.gog.com.

Nslookup returns:
Non-authoritative answer:
Name: e11072.g.akamaiedge.net
Address: 23.51.48.201
Aliases: www.gog.com
gog.com.edgekey.net

Unexpectedly:
I can access the main page in Chrome Incognito Mode.
The first page loads once I clear the cookies, but fails to load once I hit refresh (and cookies are set).

Found a reference to this issue in Akamai here:
https://stackoverflow.com/questions/26992685/an-error-occurred-while-processing-your-request

I think the site is working now? No more errors on gog and galaxy client.

Yes, it's working again!

Thanks GOG!

And the error hits back on 18 June. It was working on 17 June evening.

Exactly the same behavior - cannot load main page if cookies are set, but deep links are working.