To provide multimedia support within the Chrome / Chromium web-browser for enabling HTML5 video support, FFmpeg is hooked in at build-time. If Chrome / Chromium is not built with FFmpeg, then there is no video support. There is a problem though and that's Fedora and other Linux distributions don't ship with MPEG-4 and other select video codecs due to patent/legal worries. FFmpeg also doesn't support plug-ins in a way that would allow building FFmpeg support into the Google web-browser without these worrisome codecs. In other words, there is currently no video support on Chrome / Chromium if using packages on some platforms.
To address this issue it has been proposed that Google Chrome receives support for the G-Streamer framework. Using G-Streamer would allow multimedia codecs to be installed ex post facto, whether they be derived from FFmpeg or legally acquired through Fluendo. G-Streamer also supports a variety of hardware acceleration methods like VA-API, VDPAU, XvMC, and even DxVA for Windows users. G-Streamer is also receiving other video playback improvements thanks to a recent hack-fest.
While G-Streamer seems like a good answer and is used by a large number of software projects, Chrome developers so far are unwilling to support this open-source multimedia framework or even consider such support. Google's Andrew Scherkus has basically thrown out the idea of using G-Streamer within the web-browser since support for pluggable codecs as it's a direct conflict with Chrome's security sandbox and goals. To circumvent the sandbox/security issue, G-Streamer could be loaded when Chrome is started, but Scherkus claims there would hurt the browser's performance too much. Another issue that Scherkus has is that while G-Streamer is supported on Mac OS X and Windows too, it's not at the same level as the Linux support, which would lead Google into investing into three different media back-ends.
For now the idea of having G-Streamer support in the Google Chrome / Chromium web-browser is canned. Some though are discussing the possibility of sand-boxing G-Streamer to avoid any security issues. If you are using Fedora or any other strict free software distributions you will just be restrained from using any HTML5 video support within the browser or you must find a third-party package that is built with FFmpeg or build the browser yourself.
Follow the G-Streamer on Chrome discussion in this Google Code bug entry.