Most things are possible.
The big issue is getting rotary motion with minimal backlash. It can be done easily if you have lots of money, but not so easily if you're on a more typical budget.
Harmonic couplings are a good solution, but new they're expensive, and although there are usually plenty available second hand, they'll of generally have been replaced for a reason, so second hand ones can be a bit hit or miss.

Machine controller software wise, there are two possible options.
One, you do all motion generation with axis/tool offsets within CAM. The big benefit of this, is you can run any machine controller that can control 5 axis, but the big disadvantage is you need to setup all your axis/tool offsets in CAM.
Two, you use a machine controller that is capable of handling the axis/tool offsets. The big benefit of this is you only ever need to generate the code once, but not many controllers are capable of handling the required kinematics.

I know Dynomotions KFlop and KMotionCNC come with various Kinematic examples, which you can customise and recompile for your specific machine. TomK (the main man behind Dynomotion) seems to love this kind of thing, and is always willing to help anybody looking to customise kinematics.

I'm sure LinuxCNC can be configured/compiled for such things.
Mach3/4, I'm not sure on their level of kinematic customisation. They certainly support enough axes.


To give you some ideas of what's possible, here's some videos of 5-axis machines running KFlops-
https://www.youtube.com/watch?v=PuIdtZKdYfA
https://www.youtube.com/watch?v=XVrsO89hoV0
https://www.youtube.com/watch?v=wBrl31IrFbE

And although not 5 axis, this video demonstrates what's possible.
The hexapod takes XYZABC coordinates, and the KMotionCNC kinematic library converts that to the required motor motion.
https://www.youtube.com/watch?v=t7KMGkfuvfM
Or there is Dynomotion's 'Geppetto' 6-axis demonstration robot-
https://www.youtube.com/watch?v=MSONYPXe3bE