The guy says that they first measured the distance to another star in 1838. I want to know how much of a shock the immense number they came up with would have been?
Before the invention of the telescope, astronomers attempted to measure the annual parallax of stars but were unsuccessful. They recognized two possible explanations for the lack of detectable annual parallax: either the Earth was stationary or the stars were so far away that the annual parallax was too small to be measured. Given the accuracy with which stellar positions could be measured in antiquity, Ptolemy and the others deduced that if the solar system were heliocentric, then the nearest stars would have to be at a distance of at least a few thousand times the Earth-Sun distance.
So heliocentrism and distant stars have been tightly linked since the second century. Tycho Brahe, for instance, rejected a heliocentric model in favour of a model where the sun orbited the Earth and all the other planets orbited the Sun. He had determined that if stellar parallax existed, it was less than 1 arcminute. This would imply a distance to the nearest stars of many thousands of AU.
When James Bradley explained the aberration of starlight in 1729, this was seen as definitive proof that the heliocentric model was correct. After this point, most astronomers would have taken the lack of measurable stellar parallax to indicate the stars were very distant.
TL;DR: It was hypothesized that stars were very distant in the second century, and most astronomers would have accepted it by the 18^th.
Source: Foundations of Astrophysics, Ryden & Peterson
You might want to try r/askscience