Well, doing that with an all-purpose CPU may be possible, but certainly not easy, and with carefully hand-crafted code. Provided said CPU runs fast enough, and has appropriate interfaces. Usually the fast ADCs have some kind of LVDS I/Os, and not singled-ended ones, so you'd need to be able to interface with that, and the CPU needs to have an embedded serial controller that can handle those rates. If it has some kind of LVDS interface, for instance meant for displays/cameras, you may be able to use that to transfer ADC samples. Maybe. Something to check with some ARM SoC for instance.
That would certainly be a challenging experience, probably requiring baremetal programming, and whatever processing (short of just storing samples) you could do with that would still be extremely limited, if it's even possible.