Silly question here! However, can someone tell me their experience with which will be the most accurate 308 heavy barrel in 308. make and model please.
That is kind of a silly question because to get the "experience" one would have to have shot a dozen or two different brands/models of 308s and you're not going to find many that have done that. You also don't specify what 308 but I'm assuming you mean the 308 Winchester. In addition, you have to specify what you want it for. If you want the "most accurate" 308 you would need to go with a custom unlimited bench rest gun weighing maybe 30 lbs. on a return to battery rest but that wouldn't be too practical for most of us.
Many manufacturers give an accuracy guarantee, which in many instances is a ho hum 1 inch at 100 yards for 3 shots. Les Baer gives a 0.5 inch 100 yard accuracy guarantee for 5 shots with match grade ammo in their semiautomatic 308 Winchester and a 0.5 inch guarantee for 10 shots in their 308 Winchester bolt action rifle; I don't think it gets better than that. HS Precision gives a 0.5 inch guarantee but I don't remember if it's for 3 shots or for 5 shots.
To give you an idea of how erroneous an accuracy evaluation based on a 3 shot group can be, consider the above group shot with a Thompson/Center Encore. It's the best 3 shot group I've ever fired and it was done with a gun I couldn't get to consistently do better than 1 to 1.5 inches at 100 yards. So, to me, a gun that comes with a target showing a fantastic 3 shot group proving how good it is doesn't mean much to me.
There have been very accurate rifles made by many different manufacturers such as Remington, Ruger, Blaser, Weatherby, etc. but unless they have enough confidence in their product to give a meaningful accuracy guarantee, you may get one with an "acceptable" capability of 1 inch 100 yard groups or you could get one much better or a bit worse.