Moray if you can get away with 4 x 2 then do so because it makes the build so much simpler and cheaper because you can get away with 1 ball screw on the long axis.

Stay away from R&P it's a pain in the arse unless you use helical rack which is expensive. Even then it doesn't come close to the smoothness and efficiency of ball screws and it's much more complicated to set up and keep adjusted properly so backlash is something like acceptable. It needs a gearbox or belt n pulley ratio set up to do it correctly so get decent resolution. You don't get any of this with ball screws as you know from Mills lathes etc, just fit and forget except for odd bit of grease.

I wouldn't build a frame from anything less than 50x50x5 box and brace it up well. 5mm gives you something to tap into for the rails and lowers the resonance better than thin-wall 3mm stuff.
Stay away from the Round type rail and go with profiled rails. Much better far less hassle. Last 10x longer

Gantry use profile and My L shape design it's strong simple and proven on 100+ machines now.

Cost is difficult because as you know it's the electrics and ball screws, rails etc which eat up the money and how far do you want to go with quality and where are you willing to buy from.?
The best place by far is China but there's a lot of rubbish out there, the cheap screw/rail kits you see on eBay, etc are nearly always low quality and mismatched, like 5mm pitch for a router which you really need a minimum of 10mm. Same with the electrical kits, steppers with high inductance, and low voltage PSU.

That said if you keep it simple on the electrics and go with say Linux CNC and Parallel port with a simple BOB then it's possible to build at sensible money.