It seems some people have had some success using ubuntu16, ubuntumate or freebsd on the g5, giving you web access etc with no security issues (most seem to have some issues with 3d video card drivers, but you dont really need that for just web browsing, plus it gives you access to the full suite of linux software to boot
Apart from not using the i386 architecture, its still a quite powerful machine, I had similar issues with my old p3 machine, was quite happy running xp on the net etc etc, but simply couldnt run any newer windows. My current toy I am making up is a ultra low power net browser using a pi3/ubuntu mate that I am building into a led lcd monitor, total draw is 60w compared to 200w for the laptop or 500w plus for either of the two towers