On the issue of .303 chamber and .308 barrel bore (ie a 30-30 barrel), I will of course defer to those more experienced than I am (just about everyone on this board), however, when looking at the SAAMI chamber (minimum) specs, the differences just don't seem that significant when we consider some of the common throating issues with our Handis.
.303 chamber neck: .3400"
http://www.saami.org/PubResources/CC_Drawings/Rifle/303%20British.pdf 30-30 chamber neck: .3331"
http://www.saami.org/PubResources/CC_Drawings/Rifle/30-30%20Winchester.pdfDifference: .0069"
.303 Leade: .3125"
30-30 Leade: .3090"
Difference: .0035"
I do not have a Handi 30-30 (shame

) but my bet is that the chamber will be a smidgeon (technical industry term) larger than the SAAMI .3331" minimum at the neck and possibly even the littlest bit bigger than .309 in the leade minimum. And yet, the Handi 30-30 is widely viewed as a most excellent shooter.
In the interests of saving some $250 (the least I think a rebore could cost), would a .303 chamber cut tightly to SAAMI minspec (or just that slightest bit tighter if possible, thank you) that left those tolerances, .0035" in the leade, be worth it? Certainly for me it would. And if I found that jacketed bullets were not as minutely accurate as I desired, would I not be happy to shoot .309/.310 sized lead? I most certainly would.
Just my thoughts on the subject. As an aside, I just bought a 30-40 Krag sporter and it is now my favorite round and rifle by far. I believe that in a Handi, with judicious experimentation, the 30-40 could be loaded up to factory .308 ammo performance and that with Krag loads it will shoot 220 gr lead with 30-30 comfort. But I suppose the Handi offers the same versatility to the .303 Brit...