It wouldn’t be fair or correct to say that manned interstellar travel is impossible without giving a bit of a disclaimer. Saying that something is impossible implies that it can’t be done and never will be done. Since man has accomplished many things that were considered as being impossible, it would be safer to say that at our current level, it can’t be done. Why is that?
The biggest reason is that we are talking about enormously vast distances, so large that they boggle the mind. The nearest star to our solar system is over 4 light years away. This is the distance light would travel in 4 years. Light travels at roughly 186,000 miles every second. So to figure out the distance to that star, roughly and in miles, the figure would be 186,000 x 60 (number of seconds in a minute) x 60 (number of minutes in an hour) x 24 (number of hours in a day) x 365 (number of days in a year) x 4 (number of light years away that Proxima Centari is).
As rough and conservative as the number is, this works out to about 24,000,000,000,000 miles. Now, consider that in order for Apollo to go to the moon, it first had to achieve a speed of 25,000 miles an hour (approximately) to escape Earth’s gravitation. At this speed, a great many generations of people would live and die on the spacecraft long before they’d ever reach even the nearest star to our sun.
The question then becomes; can we travel faster? Probes we’ve sent out certainly travel much faster. However, they also don’t have people aboard who would be subjected to extremely high gravitational accelerations. High G-forces can quickly kill a person. An opposite force would need to be used to prevent this from happening.
On Star Trek, this was all done with artificial gravity and warp engines for faster than light speed. Theoretically, it might be possible, but we are nowhere near as technologically advanced, as it would require.
Ideas have been proposed wherein the craft would accelerate at a constant rate during half the trip, then flip around and slow down by the same constant rate, thus providing gravity. This shows some promise, as the maximum speed would be enormous, and the trip to Proxima Centari might only take 70 of our years, and another 70 years back, according to NASA.
There are some problems, though. We don’t have the means, currently, for maintaining a constant thrust for 35 years, nor a way to slow down for 35 years. Also, even a self sustaining ship traveling at such great speeds, encountering a speck of dust, would see the dust going completely through the ship and anyone standing there at the time.
Again, the Star ship Enterprise had deflector shields. This is theoretically possible, but we don’t have the technology right now to do it.
Keep in mind that we’ve only been talking about going to the nearest star to Sol. The distances grow far greater from there, to get to other stars.
Theoretically, it is possible to have interstellar travel. With our current level of understanding and technological abilities, it is unlikely in the extreme. Can it happen? Sure, it is possible. But it wouldn’t be a good idea to think that it would happen before our great, great, great grand kids are grown up and doing the work to figure it out. It isn’t impossible; really, we just don’t have the knowledge to do it now.