Bending space? Easy.. Just breathe and be "here"
Mass bends space regardless. The more mass, the more bent the space is which creates gravity in the first place. So we've been bending space since we've existed.
Now as far as using that capability to travel faster than the speed of light, not so much.
Now the video is funny because it uses "Star Trek" for it's example of traveling faster than light and explains why it can't be done by acceleration. However, Star Trek was about warp speed for faster than light travel. Warp speed isn't about acceleration through space, but creating infinite expansion of space behind an object while making infinite compression of space in front. The expansion is seen and can be mathematically computed, however... The compression is basically a black hole, but not since it would be inverted mass and not a collapse of mass. Otherwise negative mass to make happen.
https://en.wikipedia.org/wiki/Alcubierre_drive
It's funny because Star Trek was the inspiration for even coming up with the idea that made trying to figure out a potential way to faster than light travel, in terms of a mere academic equation, possible.