Can Science Solve Everything?

No.
As Robert Oppenheimer said, "Science is not everything, but science is very beautiful."  But having been through the usual phases of reading classic scientists, current scientists, old philosophers, new philosophers (none of the latter very interesting), and religious/theological sources, I keep coming back to the idea there are things we'll never solve.

But don't take my word for it. Here's a very good article.

Not even the most brilliant theoretical physicists can do a damn thing with the "Why" question - why is there something rather than nothing? One of the commentators on this article attempts to reason his way out of this by saying there may not be a "why" - "Why can't things just BE?" but I don't think he wins a point: if there's no Why, then we have to ask Why there's no Why. 

Of the other points, take life after death. Setting aside the much argued-over NDEs (near-death experiences), some scientists argue there's no evidence to think there's anything: certain processes in the brain stop happening, and that's the end of it.  But what evidence would there be, exactly? Sure, it would be nice if people were always coming back to tell their children where they stashed some extra cash, but there's no reason an afterlife has to be workable that way.  It's easy to postulate an afterlife state where contact with or even awareness of the physical world doesn't exist.  As a Christian, I think something does exist, but what really annoys me is that, if I'm wrong, I'll never KNOW I'm wrong.

But I digress.  The point was there are still places where science stops working.  Is there anything past those boundaries? We can debate that, but we're no longer debating science.  Science and the scientific method are the best tools ever developed to explore and understand the magnificent physical universe we are in. If God exists, I take it as a commandment that we are meant to use all the tools at our disposal to further develop our understanding and further explore (and protect) the realm we exist in. 

So let's saddle up our microscopes, our telescopes, our particle accelerators. Let us build out models of Earth's environment while we also build our starships.  This is the universe we're in. Science lets us make the most of it.