First of all, you should not install the RPM version of JRE on Fedora Core. Apparently, it causes some conflicts with Yum packages. Also, the RPM installs in /bin/java, whereas the .bin just installs under whatever directory it runs from (most typically /opt).
Get the .bin versions from Sun. Move them to /opt. Run the 64-bit installer .bin first, then rename the resulting directory in /opt. I added "-64" after "jre".
Now run the 32-bit installer, and leave the resulting directory name as-is. (Or you could rename to -32, etc.)
Create or update the symbolic link for the Firefox plugin to point to the plugin in the 32-bit JRE.
Edit /etc/profile.d/java.sh. Change PATH to point to the 64-bit version.
Use /sbin/alternatives to add 32 and 64-bit versions as alternatives for /bin/java, and select the 64-bit as default.
After doing this, Eclipse will run, and I can still use the Java plugin in Firefox.