I have the same problem, except the error code is -105
bump
If you also use systemd-resolved, then the solution for me was
ln -sf ../run/systemd/resolve/stub-resolv.conf /etc/resolv.conf (/etc/resolv.conf may be empty, I don't know why this works)
this happen to me too error code is -105 can do everything just not browse in the client
Forgot to post earlier, changing my DNS servers to google's (8.8.8.8) solved it for me.
that worked but had to restart steam
Nothing extracted yet.
I already installed all the necessary packages for arch. The launcher runs good, the Library even loads. I also tried to download a game and it worked.