I certainly could, and did when I was a kid, but that's all I had. My most accurate military rifle I own is an unmodified Springfield 03-A3 made in 1943 and it can probably stand with lots of modern rifles. But to use it in it's original format for serious hunting, no. I have too many rifles that are far better suited.
"Let us speak courteously, deal fairly, and keep ourselves armed and ready"-T. Roosevelt
I agree that the old military rifles like the enfields, springfields and mausers are still as great as ever and also that in todays world the lines have been totally blurred between military and civilian rifles. Also lets take into account that nearly every rifle designed for military use is available to civilians.
What it comes down to is what cartridge suits the purpose and what set up feels right for the guy pulling the trigger. As long as you are confident sending the round at the target, you are using the right firearm. I have several different combonations of stock style, barrel build, calibre and optics. I have loads worked up for each that do what they are asked to, but I also live by a very strict rule for everything in my safe: I will not own a firearm that is not accurate with store bought ammunition. I sold two rifles last year that could not shoot better than 1.5 MOA with any factory ammo I could find. But that is me, not you.