No-one outside the British Armed Forces is permitted a machine gun, and only then, on restricted military ranges.
You might be able to buy a handgun, but no-one other than a police officer (or government security) is allowed a working handgun/pistol. There is such a thing as a long barrel pistol that can be owned, but they're more like a short rifle.
You could own a 50 cal with membership of a range where you could use it, and as you rightly say, with a firearms license. With each bang costing over £5 a trigger pull, it would be a complete waste of money from start to finish. Plus there are better rounds for that sort of range work.
As I said, longbarrel pistol. To own one of these you would need to be a member of a gun club, likely with an indoor range of (say) 50 yards. These pistols are typically used for target competitions. Likely there would be a caveat on your license restricting the use of this to your practice venue, and recognised competitions.
Generally speaking these are 50 yard rifles used by farmers for rabbits. Beyond 50 yards their accuracy is generally pretty appalling.
308's are popular because there's plenty of cheap military ammunition around to feed the rifle, but the military stuff is loaded to higher pressures (if I remember), but it is possible to chamber the 7.62 so people use it. Personally, I think it's too much, and I don't like it, even with a sound moderator.
You get a firearms license through being a member of Home Office Approved club (and you will be restricted to using your firearms at ranges), or you have permission to shoot on land for hunting or pest control purposes. THEY are your 'good reasons'. If you have a serious criminal conviction, or done serious jail time, you are automatically banned from possessing a firearm of any sort.