While RHEL5 is old, it's still Red Hat and it's extremely popular. You should find a binary package for just about any software you'll want. Might not be the latest version, but you'll find everything.
So I would avoid compiling stuff as much as necessary. ask your admins firmly to do their job and install stuff you need from official repos, and then ask them nicely if they can install stuff from EPEL repo and other 3rd party repository. THIS IS THE BEST OPTION.
If they won't install stuff from 3rd party repository, just download the .rpm files yourself, and extract them under your home directory. With LD_LIBRARY_PATH environment variable you should be able to do anything you want. It won't be a pleasant experience, you'll run into a dependency hell of downloading rpm packages, but I think it's still better than compiling everything you need. This is how I manage at my university's servers... again, not pleasant, but it works.
Things I haven't tried but could work better:
http://0install.net/injector-feeds.html (I really should have started with this, sorry) - heard about it a long time ago, didn't have the packages I needed. See if the packages you need are here.
https://github.com/dex4er/fakechroot/wiki - you might be able to install your own linux distribution with this. kinda like a virtual machine only using system resources directly. Haven't tried this at all, don't know if what I suggested is even possible, but it's a possible direction to check out. if you have a disk quota, probably not a good solution.
Take a look at the "success" link that /u/Drupyog shared; there are plenty of GUIs there.
Also, 0install recently ported from Python to Ocaml, and it has a cross-platform GUI. I really recommend reading Thomas Leonard's blog posts on how he selected Ocaml for the 0install rewrite, and wrote the new implementation.
> As it stands right now, I can't even properly distribute Linux binaries because the version of glibc on my machine is too new
I did a quick search and found RHEL/CentOS 5 uses glibc 2.5 which is older than Ubuntu 10.04 and Debian old stable (squeeze). So, if you don't have any other dependencies it is mostly safe to use it.
You can also take a look into http://0install.net/
A cross-distribution application installer with shared libraries, digital signatures for packages that uses distribution packages for dependencies when possible is already available - take a look at 0install.
It'd be nice if Linux had a proper cross-distro packaging solution so that NVIDIA could release one repository that could then be added to ALL distros. That'd give us all more freedom by empowering even the little distros that would never see a major piece of software implementing a PPA for them. Standards FTW. Why hasn't this been done yet? I've always figured it was because the big distros don't want a solution like this because they couldn't use their massive repositories of packages to attract users.
The only solution I know of is still Zero Install, but it's not integrated with existing package managers and instead requires adding an additional package manager.