This article from New Scientist overviews several of the things which we haven't been able to explain. None of these questions are unanswerable, but a few probably require experiments so complex and expensive that it will be centuries before we get to the bottom of things. Others we might figure out in a decade. But that's science: it never stops.
- The placebo effect: how can you trick the brain into relieving pain?
- The horizon problem: why is the universe so uniform in temperature over billions of light-years?
- Ultra-energetic cosmic rays: some are more powerful than theory predicts unless they come from nearby, but there are no known sources.
- Belfast homeopathy results: related to the placebo effect?
- Dark matter: what the hell is it?
- Viking's methane: a positive but unconfirmed test for life on Mars from 1976.
- Tetraneutrons: they should be impossible, but have been observed.
- The Pioneer anomaly: something is subtly tugging two of our most distant spacecraft off course.
- Dark energy: again, what the hell is it?
- The Kuiper cliff: is there a tenth planet?
- The Wow signal: there's either something unexplained happening in space or we picked up an alien transmission in 1977.
- Not-so-constant constants: the laws of physics themselves may have changed over time.
- Cold fusion: can we do it or not?
Sometimes people say that science requires faith just like religion. These people are full of shit. The faith required to believe in God is the faith that you don't need evidence to know things. One might say that is in fact the definition of faith. Science is based on the opposite premise: there must be evidence to base any belief upon. More importantly, scientists admit when they don't have the answers.