In the most literal case, the answer to your apparent question (which is not terminated by an interrogation point but is otherwise phrased much like a question) is "No"... because no such thing as a "CNC" has ever existed in the sense you mean. "CNC" is an adjective, not a noun. Therefore, it is not possible to "build a CNC" at all.
Opening up our translation a little and assuming that by "a CNC" you really mean a CNC router or laser engraver or 3D printer or sewing machine... yes, it is possible to build such a machine using a linear motion system which does not contain any timing pulley or lead screw. You never asked how to accomplish that, but there are many ways of getting it done.
In response to your stated main objective... the cheapest method would be to manually drag you axis back and forth with tiedown twine stolen from Home Depot. That probably isn't going to feel very satisfactory to you, though. I think what you're actually looking for is going to be a stick of All-Thread rod and some clanky nuts that fit it, maybe even some shopmade ones so you don't have to weld them (or glue them) to your carriage. I wouldn't expect your accuracy and repeatability to be any great shakes, but it would do the job very cheaply.
Given that you'll likely want greater accuracy than that, you'd best get used to the fact that any reasonably accurate positioning system is going to be a bit more expensive than you're ready for. That's a good thing, since it means that every dog's man isn't wasting his time & money building a nasty CNC machine in a back bedroom with baling wire and bubble gum.
Just sigh and relax and surf Amazon and eBay looking for Asian import lead screws and ballscrews. They'll get you there. The lead screws will be the most accurate, the timing belts will be fastest, the ball screws will fit somewhere in the middle between speed and accuracy. Oh, they'll also wear worse than the lead screws but not as bad as the timing belts.