I filed a bug here: https://bugzilla.rpmfusion.org/show_bug.cgi?id=3046 (it will warn you about rpmfusion bugzilla's self-signed certificate)... but it turns out that this is not specific to the rpmfusion package, it also happens with your official tarballs in http://repo.steampowered.com/steam/pool/steam/s/steam/

Extracted steam on Fedora 20 x64, launched it, logged in with my steam ID, then after a few seconds I get an error dialog instead of the steam main window:

Could not connect to Steam network.
This could be due to a problem with your Internet connection, or with the
Steam network. Please visit www.steampowered.com for info.

Of course, I ruled out the possibility of a firewall/NAT/etc. interfering (besides, a Windows machine on the same network has no issues) and the Steam servers are online.

I also checked ~/.steam/registry.vdf and "OfflineAFS" == 0, and "Offline" == 0.

Here's what I get in the terminal:

$ ./steam
Running Steam on fedora 20 64-bit
STEAM_RUNTIME is enabled automatically
Installing breakpad exception handler for appid(steam)/version(1361807486_client)
Installing breakpad exception handler for appid(steam)/version(1361807486_client)
unlinked 0 orphaned pipes
Gtk-Message: Failed to load module "pk-gtk-module"
Gtk-Message: Failed to load module "gail"
Gtk-Message: Failed to load module "atk-bridge"
Installing breakpad exception handler for appid(steam)/version(1361807486_client)
Fontconfig error: "/etc/fonts/conf.d/10-scale-bitmap-fonts.conf", line 70: non-double matrix element
Fontconfig error: "/etc/fonts/conf.d/10-scale-bitmap-fonts.conf", line 70: non-double matrix element
Fontconfig warning: "/etc/fonts/conf.d/10-scale-bitmap-fonts.conf", line 78: saw unknown, expected number
Fontconfig warning: "/etc/fonts/conf.d/65-0-lohit-bengali.conf", line 32: unknown element "langset"
[1216/160014:WARNING:proxy_service.cc(646)] PAC support disabled because there is no system implementation
Installing breakpad exception handler for appid(steam)/version(1361807486_client)
Installing breakpad exception handler for appid(steam)/version(1361807486_client)
Installing breakpad exception handler for appid(steam)/version(1361807486_client)
Installing breakpad exception handler for appid(steam)/version(1361807486_client)
Generating new string page texture 2: 48x256, total string texture memory is 49,15 KB
Generating new string page texture 3: 256x256, total string texture memory is 311,30 KB
Installing breakpad exception handler for appid(steam)/version(1361807486_client)
Installing breakpad exception handler for appid(steam)/version(1361807486_client)
Installing breakpad exception handler for appid(steam)/version(1361807486_client)
Adding license for package 0
Adding license for package 34
Adding license for package 7802
Adding license for package 11072
Adding license for package 12361
Adding license for package 12517
Adding license for package 27399
local (potentially out of sync) copy of roaming config loaded - 1797 bytes.
Generating new string page texture 4: 128x256, total string texture memory is 442,37 KB
Generating new string page texture 5: 512x256, total string texture memory is 966,66 KB
Generating new string page texture 6: 24x256, total string texture memory is 991,23 KB
(steam:11191): LIBDBUSMENU-GLIB-WARNING **: Trying to remove a child that doesn't believe we're it's parent.
(steam:11191): LIBDBUSMENU-GLIB-WARNING **: Trying to remove a child that doesn't believe we're it's parent.
(steam:11191): LIBDBUSMENU-GLIB-WARNING **: Trying to remove a child that doesn't believe we're it's parent.
(steam:11191): LIBDBUSMENU-GLIB-WARNING **: Trying to remove a child that doesn't believe we're it's parent.
(steam:11191): LIBDBUSMENU-GLIB-WARNING **: Trying to remove a child that doesn't believe we're it's parent.
(steam:11191): LIBDBUSMENU-GLIB-WARNING **: Trying to remove a child that doesn't believe we're it's parent.
(steam:11191): LIBDBUSMENU-GLIB-WARNING **: Trying to remove a child that doesn't believe we're it's parent.
(steam:11191): LIBDBUSMENU-GLIB-WARNING **: Trying to remove a child that doesn't believe we're it's parent.
(steam:11191): LIBDBUSMENU-GLIB-WARNING **: Trying to remove a child that doesn't believe we're it's parent.
(steam:11191): LIBDBUSMENU-GLIB-WARNING **: Trying to remove a child that doesn't believe we're it's parent.
(steam:11191): LIBDBUSMENU-GLIB-WARNING **: Trying to remove a child that doesn't believe we're it's parent.
(steam:11191): LIBDBUSMENU-GLIB-WARNING **: Trying to remove a child that doesn't believe we're it's parent.
(steam:11191): LIBDBUSMENU-GLIB-WARNING **: Trying to remove a child that doesn't believe we're it's parent.
(steam:11191): LIBDBUSMENU-GLIB-WARNING **: Trying to remove a child that doesn't believe we're it's parent.
(steam:11191): LIBDBUSMENU-GLIB-WARNING **: Trying to remove a child that doesn't believe we're it's parent.
(steam:11191): LIBDBUSMENU-GLIB-WARNING **: Trying to remove a child that doesn't believe we're it's parent.
unlinked 2 orphaned pipes
CAsyncIOManager: 0 threads terminating.  0 reads, 0 writes, 0 deferrals.
CAsyncIOManager: 140559 single object sleeps, 0 multi object sleeps
CAsyncIOManager: 0 single object alertable sleeps, 2 multi object alertable sleeps
[2013-12-16 16:00:13] Verifying installation...
Shutting down. . .