Once, a long time ago (1992 or so), I went through something similar to this when I bought a SCO system with no development environment. Bootstrapping it up to having a full development environment was a gigantic pain, and not at all easy. Having library header files or gcc on a system would make your job a whole lot easier.
It depends a lot on just how obnoxious the library has been about what kinds of things are installed. If there is no gcc there, your job becomes a bit harder. If there are no header files for glibc there, your job is a LOT harder.
Also, do you get an account on the system so you have a home folder that's consistent from login to login?
If you have no gcc there, you need to find a pre-compiled binary of gcc/g++ and install it somewhere. If you have no header files there, you need to find copies of those and put them on the system.
There is no 'standard' way of installing gcc in your home folder though. All of the solutions are going to have some manner of hand-rolling involved.
Have you asked the librarians if they can change what's installed because you want to learn a bit of programming and only have access to their computers to do it with? That might well be the easiest solution.
From your comment it seems that you do have gcc and if you can compile C code, you have the library header files. So now it's a matter of actually compiling your own version of g++. You could probably find a way to entice the package manager on the system into installing a binary package somewhere other than in a system folder. I think this solution is less fun than compiling your own, and I think there may also be possible subtle problems as that installed package may be expecting to find things in particular places and not finding them there.
First thing to do is to make sure you've downloaded the right source for the gcc package. The place to find that is the GNU United States mirror page. You want to find the gcc-4.5.0.tar.bz2 or gcc-4.5.0.tar.gz file on the mirror site you choose. It will likely be in a gcc directory, and a gcc-4.5.0 sub-folder.
After you have that downloaded, you should untar it. In general you shouldn't build gcc in the folder you untar it into. So create another sibling folder that you actually want to build it in labeled gcc-build. Then the command you want is ../gcc-4.5.0/configure --prefix=$HOME/.local --enable-languages='c c++'.
gcc does require some other packages be installed in order to be able to compile itself. You can use the same --prefix line for these packages to install them in the same place. The gcc website has a list of pre-requisite packages.
$HOME/.local is sort of the standard accepted place for per-user installs of things.