Fedora won't distribute from their site because Macromedia's license doesn't permit it, but there are no other legal barriers to using the RPMs at http://macromedia.mplug.org/.
Follow those directions, dropping the Macromedia repository configuration in your /etc/yum.repos.d/ directory. It should look much like this:
[macromedia] name=Macromedia for i386 Linux baseurl=http://macromedia.mplug.org/rpm/ enabled=1 gpgcheck=1 gpgkey=http://macromedia.mplug.org/FEDORA-GPG-KEY
Then do the installation:
yum install flash-plugin
Installing this RPM should put the plugin in your Firefox plugin directories (and Mozilla's as well).
You can test your flash support at the official test page. Note that you may have to kill and restart your browser after installing the plugin; I tried the Flash test immediately and it crashed Firefox.
There's a gotcha: The Macromedia plugin works only for 32-bit Intel boxes. It completely fails on an x86_64 running in 64-bit mode (in general, 32-bit plugins won't work in a 64-bit browser). Currently there are three projects addressung this problem:
gplflash1 is the orignal GPL Flash plugin. Only handles SWF up to level 4.
gplflash2 half-works, with painful amounts of flicker, and frequently crashes Firefox. This rewrite of gplflash has been abandoned in favor of gnash.
gnash promises full support for SWF up to level 7, but is still in early development and surrounded by warnings.
I think the right stopgap on 64-bit systems might be to build and install gplflash1, but as of 21 Mar 2006 I can't because it needs an X header file I cannot yet find the right RPM for. This may change as the dust settles around FC5.