I'm applying the technology available to Star Trek to our likely reality as a space-faring race. I am ruling out the possibility that the Universe works like weekly episodes of fiction. The biggest concerns for us with space travel are distance, time, radiation, micrometeors, atrophy due to lack of gravity, medical issues, the psychology of small quarters among few companions, and harsh destinations.
As others noted, we do not know what is out there, but even in your nod to psychological factors, some of the same
shipboard threats seen on TOS could lead to death aboard a ship 300 years in our future, such as mass insanity caused by phenomena unknown to early 21st century science.
Who is to say the dangerous behavior seen in
"The Naked Time" or
"The Tholian Web" would not have real world parallels in the space-faring centuries to come?
The Universe is almost certainly teeming with life, but if there were evil empires or aggressive aliens in the neighborhood, we would likely have heard about it or been conquered by now. And those evil empires could be a millions or billions of years in the past or future and we are alone to rule our space.
...or, as explored in numerous sci-fi stories, perhaps they have watched us all along, but find us too uninteresting and/or inferior to contact right now. If you grant that the universe is teeming with life (at present), then imagine what humankind will face 2 to 3 centuries from now? There's no way to imagine a universe where some kind of external threat would not take its toll on space travelling humans. Technology (which will be common / standard in its age) will not provide 100% protection from risk, or death, unless man has moved to the point of becoming God-like (think Sargon).