As far as I can tell, anyone can enlist and become a non-com crewman, but according to a Voyager episode ("Meld" I think), non-coms can be recommended for the officer program. So, already it's working differently than the actual military. Then again, I know less about the actual military than I thought I did, so I could be wrong on that.
By the modern military I guess you mean the United States Military as other nations have their own standards. Both NonComs and enlisted lower ranks can be recommended for officer training, but a recommendation alone is not enough to get a service member into an academy a ROTC program or Officer's Candidate school.
Maybe to be an officer, you need good grades, but any schmuck can become a non-com. Or maybe to be a officer, you just need to take a lot more schooling. Non-coms can clearly advance to become officers at some point, just like in the real military (okay, I know for a fact that happens, at least).
Any enlightenment on this issue? I realize this sort of thing is poorly developed in snippets across the entire breadth of the franchise, so we don't have much to go on.
There is no evidence that any schmuck can be a NCO, in fact schmuck status seems to be reserved for your basic Ensign and Lieutenant before the latest movie. As the only one of his kind we have extensive screen time with Chief O'Brian come across more like the Top Soldier, who has been more rigorously selected then the young Ensigns and Lieutenants assigned to his department on DS9