How hot is too hot.

[fishinguy]

When Im at the range shooting my 30-06 handi the barrel gets hot really quick.  After about 4 shots its hot enough that I dont want to hold my hand on it.  The accuracy does not seem to deminsh if I continue to shoot... so I would like to know that if i continue to shoot will it hurt anything?  Wheres should be the cut off point?  I currently do 3 shots and let it cool for 5-10 minutes.

[Dillohide]

My Handi 30-06 has not come in yet     but I regularly shoot my 70's vintage Winchester Model 70 bolt action in 30-06 with a tapered barrel using 165 Hornady BT SP with just under a max load of Varget. The barrel does get hot quick. I also shoot 3 rounds, cool about 5 minutes, run a bore mop and go again. Most people, me included, cannot handle more than about 20 rounds of 30-06 recoil at one sitting so you should be fine. However, if you cannot put your hand on the barrel and hold it there for 2 or 3 seconds then it's to hot and you need to wait until you can. Leave your action open during cool down, it'll cool quicker.

[myarmor]

... so I would like to know that if i continue to shoot will it hurt anything?  Wheres should be the cut off point?  I currently do 3 shots and let it cool for 5-10 minutes.

I personally have never had the distinction of actually shooting out a barrel completely. Though I have shot enough to make for throat errosion/wear.
Your practice of 3 shots and let cool method is a good one.
Keep it up and you will have a good condition barrel for a long while. These Handi barrels are of good quality, and from my stand point the more I shoot them the better broken in they become and the better accuracy potiential they release.
Just keep up Regular and Proper cleaning.
Shoot and Enjoy your rifle, it certainly sounds like you do. 
-Aaron