How to uncompress a 9GB file in Windows FAT32

There is no way you can go over the 4GB limit of the FAT32 Filesystem, because FAT32 imposes this limitation by design. This file system has a natural limitation for the maximum file size it can handle, which is 4GB.

And yes, I haven't found any tool that splits the video file while unzipping it.

The solution to this if you do NOT want to convert to NTFS is:

• Get an external drive with NTFS filesystem on it.

• Extract your Video file there.

• Then use a video splitting software to split it in multiple parts, and then copy those to your FAT32 filesystem.

You can try to play the RAR (sic!) file with VLC media player. It does have internal support for RAR. This would only work if your RAR is not encrypted. I do not know the internals of VLC's implementation but I think VLC does not create temporary files, then this might work.
EDIT2: will not work with compressed archives, but should work with multi-volume RARs.

EDIT:
But if you want access the real file and playing it is not enough: Mount the RAR file with WinMount http://www.winmount.com as a virtual drive.
EDIT2: Does not work, stops after about 4GB (tested by asker)

EDIT3:
OK I just remebered that Microsoft Virtual PC can do multi-volume VHDs on FAT32. Related: http://www.k2underground.com/blogs/blacktop/archive/2009/06/26/microsoft-virtual-pc-and-fat32-disks.aspx
Just keep in mind that you cannot use these VHDs without MS Virtual PC. To use or open them with other software they have to be joined first. Its a little overkill just to open a file but you might try it. I have however not tested this on my new Windows 7 PC only on my older PC with WinXP.

You can boot a linux live distribution and mount your file system. Then run

7z -so e file.rar | split -b 3500M

That way you end up with the split file video on your FAT partition. I have no clue if any player is able to actually display the file.

10 min. Screencast? It's probably badly encoded (or maybe even not at all).

Why not reencode while unzipping? It won't require external drive. For 10 min. you could possibly end up with a file of around 100mb in size.

Assuming Windows Command Line you could try:

unrar.exe p -inul bigCast.rar |   ^
 ffmpeg -i -                      ^
 -acodec libvorbis -ab 128k -ac 2 ^
 -vcodec libx264                  ^
 -preset slow -crf 22             ^
 -threads 0 smallCast.mkv 

The unrar p instructs rar to print the content to STDOUT. Than it's read back by FFmpeg -i -. The ^ characters are line break escape characters from Windows .bat file format.

You can get FFmpeg from Zeranoe FFmpeg Builds site. For simplicity pick the static one.