I'm thinking of building a 300 H&H; would a #3 (.625 muzzle) or a #4 (.650 muzzle) be better for a barrel contour?

Different barrel manufacturers have different dimensions for their barrels. One maker's #3 may be heavier than another maker's #4. You'll need to say what manufacturer so we know what you're talking about. As a rule though, I'd say go with a #3. I once built a 7mm STW with a #4 Hart barrel and regretted it, too heavy. My 300 WSM with a #3 Hart is about right, and it wouldn't bother me if it was a bit lighter.

I would raher have the heavier barrel.

knowing that all manufactures have different contours for the numbers, i always understood that a #3 was more or less a "standard sporter" contour, found on most of the guns on the shelf and pretty much the same from maker to maker. at least to the point that when one says ," a #3 contour", it's usually understood to mean about a .625 muzzle dia....more or less an industry standard.
with that idea, i've always used it to mentally compare one contour to another when looking at barrels. is this correct? ....i guess i really never looked close enough to different makers contour numbers to proove this out.

I'd be using a Shilen barrel, finished up at 25"

This is a hunting rifle, right? The lighter contour will make a more portable gun. There is no reason it can't be every bit as accurate as the heavier barrel for the first three or four shots before heating up -- after that a loss of a quarter-inch of accuracy is hardly a significant issue, assuming you aren't varmint hunting with it

The Shilen #4 is called Mag contour use to be 28" finished length was 26" and the #3 Sporter Contour 26" finish 24".

The #4 has longer chamber length you may want that with the 300 H&H.

I've always erred on the side of heavy, and for barrel length, I've erred on the side of longer.
As an example, my recent 257 Roberts build had a number 4 barrel and finished at 26" length.

Don

DMB,
you can't go wrong there, it's not that much harder to carry around an extra 1/2 lb. or so and the roberts can use the length with 120 gr. bullets.
i went with a #4 on my 6.5x270AI also. it made a gun that's just a tiny bit heavy up front, but it sits very steady off-hand because of it.
my gunsmith built himself a 7x57AI with all the same components as my 6.5 right before he built my gun and after shooting his, i told him to build mine the same way.