Any way to update OSX 10.4's Bash version?
Do you have gcc and other tools installed?  You can check with which gcc and then gcc --version.  If you don't, you can install these from the Developers Tools DVD that came with the system, or you can download from Apple or gnu, but it might be tricky to get a version that will run.

Once you have gcc and the other command line tools installed, the next challenge is to obtain a bash source that will successfully compile and build on an old OS.  I'm going to try this myself for 10.5 and 10.6.

Note that you don't need to worry about having a vulnerable Bash if the machine doesn't run a web server and isn't running services exposed to anyone but you.  If you're the only user, and you don't run web sharing or other external services, you don't need to both.