Remmington 03A3 sniper rifle?

[vintage hunter]

The local gun shops here are saying it's only worth $250 $300 as it is

That gunshop is blowing smoke hoping to con you out of your rifle. An original un-messed around with 1903a4 sniper bolt alone is worth more than that. The correct stock turns up once in a while on ebay. You shouldn't have any trouble finding the rest of the parts there as well. If your planning to go whole hog with the restoration your going to need a set of rings also, what you have now are 1 inch commercial rings from the 60's or 70's. They sometimes turn up on ebay, and the correct scopes too.

I mean if the gun is worth only $500 restored

It's worth more than that the way it is, and it will be worth considerably more afterwards provided you use all correct/original parts in the restoration. It's not going to be a cheap undertaking but with a bit of patience it can be kept reasonable. I'm not really up on what 1903a4 parts are going for at the moment but last account I had stocks were in the $300-$400 range, a correct scope about $1000 or so and rings probably $150-$200. The bands, screws, swivels, hand guard ect. are standard 1903a3 parts and not all that expensive, I figure another $150 or so. These are auction prices and you may be able do a little better by dealing with a individual rather than competing with the multitudes on the auction sites.
Also you mention the barrel is chrome lined. I may well be wrong but I've never heard of any original 03a3/a4 barrels being chrome lined.