computers are modular, rifles are not. The best rifles take a lot of hand-fitting.
I suppose to some degree AR-15s could be considered modular, but beyond that you really must be a gunsmith with a good set of tools and some special know-how to start with a bunch of parts and end up successfully assembling them into a finished product
(note, reassembly of an exisiting rifle is a fair amount easier)
Let's talk parts for a minute. The heart of the rifle is the recievier. It is also the part that has the serial number. As far as the law is concerned, this simple part IS THE GUN, so you will still need to buy that from a FFL.
The parts that are available are more designed as replacement parts to fix something that is broken, or upgrade parts. If you did have a reciever and had the skills and tools, it would be MUCH more expensive to build a firearm from the ground up.
So, what are your options?
Buy used! Check out Gunbroker.com, they are like ebay for guns.
Buy a single shot H&R, NEF, Rossi, or Remingtion 'Spartan International' break-open rifle
http://www.budsgunshop.com/catalog/product_info.php/products_id/13126
http://www.budsgunshop.com/catalog/product_info.php/products_id/41856
http://www.budsgunshop.com/catalog/product_info.php/products_id/20739
http://www.budsgunshop.com/catalog/product_info.php/products_id/55048
If you can scrape together a few more dollars, you can get the bottom rung of bolt action rifle, which are really pretty solid guns
Stevens IMHO is a real good buy for the money
http://www.budsgunshop.com/catalog/product_info.php/products_id/14605
http://www.budsgunshop.com/catalog/product_info.php/products_id/57451
There are pre-scoped options, but the scope is pretty cheap, If your budget is tight it might be okay for a year or so, then after that plan on upgrading to a $100 scope, if not a $200 scope.
Yes, it is quite reasonable to put a $200 scope on a $300 rifle.
Remington 715 and 770 are okay, but if oyu come across a Remington 710, RUN AWAY. I think the Remington 770 is superior to the 715, but so far I have only seen them packaged with a scope. Same as above, the packaged scope is pretty darn cheap, don't expect it to last more than one season.
http://www.budsgunshop.com/catalog/product_info.php/products_id/41972
http://www.budsgunshop.com/catalog/product_info.php/products_id/42071
The last two to consider as low budget rifles are the Marlin X and the Mossberg 100
http://www.budsgunshop.com/catalog/product_info.php/products_id/57399
http://www.budsgunshop.com/catalog/product_info.php/products_id/43850
They too have 'scope included' packages.
The 243 is a wise caliber choice