The Bible records many miracles, direct acts of God that are without natural cause. They are not explainable by natural laws. There is no natural “cause and effect” happening in a miracle. That's what "supernatural" means, beyond nature.
But hard science involves the observation of natural cause and effect and the ability to predict an outcome based on its findings. It is entirely based on observing and drawing conclusions from the natural world.
If we diagrammed these two, we would have miracles in one circle and cause and effect in another with no overlap.
That does not mean that both do not or cannot coexist. Not at all. Science, by its very nature--limiting itself to natural causes, could not see a miracle right in front of it because there would be no natural cause to observe and it would assume, "I must be missing the (natural) cause." It’s the wrong tool to see a miracle. It’s like using a microscope to study a distant star.
Ethics and values lie outside the realm of hard science. They are real, but hard science does not make these kind of value or ethical judgments. Science is what, when, where, how, how much and why (cause/effect). But we live in a world that has ought, should, promise, fair, and many more words you could add, none of which hard science touches on. These are real, yet lie beyond the bounds of the scientific method.
History itself cannot be directly studied by hard science. Yet history is real. It happened. However, you can’t put George Washington crossing the Delaware in a test tube and repeat it with controls. We cannot observe it, unless we invented a time machine! Instead we can search for evidence to make a case for or against it, a forensic case.
But evidence, though admissible in a court of law, is not the same as scientific proof with mathematical certainty. True, DNA can be matched to a high degree of mathematical certainty, but we can't strictly "scientifically" prove how the DNA got there at the crime scene. That's history! All we can do is use reason and artifacts to make a forensic case. And we can be "beyond a reasonable doubt" based on evidence.
It is just such forensic evidence we have for miracles such as the Resurrection of Christ. Even though miracles aren't caused by natural causes, they do leave a trail, evidence, even eye witnesses.
We do have evidence and eyewitness testimony for the miracles of Scripture. In The Case for Christ, Lee Strobels, a former skeptic, lays out the historical evidence that convinced him the Jesus really did rise from the dead.
Often there is are philosophical presuppositions that, for some, rules out the possibility of anything supernatural from the start: “Miracles can’t happen.” Why not? The necessity of that position cannot be proved. It is a choice. It is actually a faith position! And as long as we are all honest about it, that's great.
Finally, we should note that a miracle is not really "violating" a natural law any more than someone catching a falling ball violates the law of gravity. It is overriding it by entering into it's frame of reference, regardless of whether we are talking a natural cause of supernatural one.
So, can a person be scientifically astute and also believe in miracles? Many do! In the book The Case for Faith, Lee Strobels interviews a number of leading thinkers, scholars, and scientists who are Christians. We recommend this book for more on this subject, and it’s companion, The Case for Christ.
What about the many fields of study also called "science" such as social science, psychology, environmental science, geology, and more, These all blend some hard science with opinion, reasoning, probability, cataloguing, historical research, polling, or even philosophical positions. All of these are really making forensic cases ultimately for their conclusions, even if factual findings are used at many points. As such they present evidence which must be weighed.
You can't put George Washington crossing the Delaware in a test tube. For past events, we must rely on evidence and witnesses.