Basically you use road components. with different tires and brakes.
Fortunately for the racer/rider on the budget, the cross scene is very tolerant of frankenbuilds.
Some cost-saving options:
-Singlespeed. You can ad gears later, but this is a cheap way to get into it. You can get a normal entry-level road wheelset from Nashbar/Performance/etcetera and then a nashbar singlespeed kit for about 20 bucks. This would be my route of choice.
-Bar-end shifters.
-1 x 9 build, possible with a single barcon to shift the rear cluster. There are lots of fancy chain guard options, but an old front derailleur will work.
-Old cantilver brakes that were high quality in their day (XT/XTR) go for a song on ebay a lot fo the time.