"Rejecting" standard real analysis is (usually) not rejecting that real analysis's theorems can be derived from the axioms used.
Rather, it is rejecting it as a reasonable model of numbers.
When you take axiomatic mathematics and you run with it, you end up with strange results. Things like Gödel numbering lets you show that your axiomatic system for even something as simple as addition, subtraction, multiplication and division isn't able to exclusively model what you'd intuitively think was numbers.
To evaluate an axiom system, we have to look at what it is used for. Sure, this is icky applied mathematics, but if our axiom system of counting numbers (for example) doesn't model what you do when you go "one chicken, two chicken, three chickens, ..., 5,272,992 chickens, ...", maybe you should reconsider your axiom system for counting numbers.
People who "challenge" or "don't accept" standard real analysis think that alternative sets of assumptions -- axioms, rules of proof, etc -- produce better or maybe just otherwise useful results.
One example is constructive real analysis. Here, we start with pretty much the same assumptions, but we remove the law of excluded middle -- that a statement can be assumed to be either true or false. We still claim that no statement can be both. There just isn't a rule that goes "!!X implies X". There is still a rule that "!!!X implies !X", which can be derived from the other axioms of logic.
There are some other subtle changes as well.
That, plus being a bit careful about what other axioms we use, and changes to some of the definitions of terms in analysis (these "redefinitions" can be shown to be equivalent in standard set theory to the standard definitions usually), gives us an interesting property; that you can take any proof of existence of a object and mechanically turn the proof into an algorithm that produces the object.
So if you have a proof that says "there exists an X with property P(X)", you can always write out the digits of X (well, the algorithm to do so might be expensive).
There are other non-standard real analysis as well. Some permit infinitesimals -- values that are bigger than 0, but smaller than any number you can write down -- and do calculus where dx/dy can be calculated by doing infinitesimal mathematics.
We call all of these a form of "real analysis", because at the scale of doing calculus in the service of physics, these all end up agreeing. They can all agree and derive that a car with an acceleration of a over time t travels 1/2 a t^2 distance, they call can produce their equivalent of the fundamental theorem of calculus, etc.
Sometimes there will be slight differences. For example, the intermediate value theorem states that any continuous function of one variable that starts above a line
and ends below a line crosses the line. The constructive version instead concludes that it gets within any arbitrary distance of the line.
Because there is no effective procedure that permits you to take an arbitrary continuous function, a proof that it is above the line at one point, and below at the other, and produce a decimal expansion (or equivalent) of the location where it crosses the line ... constructive analysis doesn't give it to you.
Constructive analysis does give you a sequence of points $p_i$ such that $|f(p_i) -k|$ converges to 0, and the points $p_i$ all lie within a closed interval; in classical analysis, this guarantees a convergent subsequence. In constructive analysis, this doesn't guarantee a convergent subsequence, because there is no way to find that convergent subsequence!
No physical experiment could distinguish between these two claims, because they disagree in the limit. So both model reality. One just models a reality with additional, untestable claims (and that is the classical analysis version).
One can find nonstandard analysis useful without "rejecting" standard analysis. As an example, when you are doing geometry on a computer, being aware of constructive analysis theorems and their difference with classical analysis can help illuminate some things you shouldn't assume.
And this isn't just games. A recent paper - popular article - uses intuitionalist/constructive reals and logic to describe a non-time-symmetrical general relativity universe. Because a fully time-symmetrical universe requires infinitely dense information at the big bang; sort of like an infinitely precise real number.