I don't really understand why you have such unusual constraints. No
installation, no live CD and low network traffic excludes the obvious
solutions like booting a distro from USB, setting up a VM or using a remote
system via SSH.
How do you actually plan to run such a system? If you really only have a
browser check out the JavaScript qemu port.
But I doubt that you have enough resources to cross compile applications or
that you will be able to use such a system efficiently.
- free
Most of the linux distributions are free.
- gcc, binutils, bash
Most of the distributions ship with those apps either pre-installed or
installable via packet manager or from source.
- low network traffic e.g. =< 1kbps
Linux distributions don't generate network traffic. It's the applications that
generate the traffic. But the problem here probably is that I don't understand
what you want to do.
- sufficient resources to cross-compile gcc
That's the crucial point here. The JavaScript solution does not provide
sufficient resources and you are not allowed to install a distribution
locally.
- ability to install programs from repos
Possible with most of the available distributions.
Depending on what you actually want to do, I guess the best way is to set up a
remote server and using an SSH solution which allows access from a browser
(e.g. via Java applet). The network traffic is low and you don't need to
install additional software.