I mean, let's take an example of science. Is there have any theory that was contradictory in nature but proved correct in observations or practical?
Consider the 'Grandfather Paradox', and Relativity. Closed Timelike Curves seem to be possible according to the math of Relativity, but we don't see evidence of chronology violations - so much so that we have the Chronology Protection Conhecture. Discussed in more detail here Paradoxes never exist in nature, so why does the grandfather's paradox make sense in physics?
Contradictions and paradoxes really show contradictions between inferences of our premises, and that can mean gaps where an otherwise good theory fails - and identify areas and phenomena for new scientific work. In the case of Relativity we think it is not a complete theory, and other apparent paradoxes about black holes also seem to be accounted for by this, naked singularities and cosmic censorship hypothesis, blackhole information paradox and Hawking Radiation.
What If theory can be proved practically correct even if being contradictory to itself in theory? Can it be possible?
No, this is not possible.
Theories are checked against the facts of observation. Most precisely, you compare to observations whatever you can infer logically from the axioms of the theory.
The impossibility here comes from your assumption that the theory is self-contradictory, which I have to guess means that some two axioms are contradictory to each other. It is a basic principle of deductive logic that contradictions are false, so the conjunction of all the axioms of your theory will be false because it contains a contradiction.
Given this, it is not possible to infer anything from the axioms taken together.
You could draw inferences from various subsets of the axioms not containing the two mutually contradictory axioms but this would be fallacious. Whatever conclusion you infer logically has to be consistent with all premises, and this is not possible if any two premises are mutually contradictory.