Radar doesn’t work in Astronomy, beyond the distance of the moon or perhaps Venus - because the beam becomes so faint as it spreads out that it just doesn’t have enough “brightness” to see the reflection. We know our own distance from the sun through careful radar measurements of our distance to Venus and by observation of Venus as it goes behind the sun (known as an occultation).
So, how do we tell interstellar distance?
It’s a fascinating story.
For relatively nearby objects, like stars out to about few hundred light years (which has been updated to about 10,000 light years with more accurate measuring tools), we use parallax. How much does the object shift against the background as a result of our orbit around the sun. Look at a star relative to the rest of them, and we can see some that move relative to that background as we swing to the other side of the sun. Our orbit is about 92 million miles, which astronomers call an “Astronomical Unit” or AU. We choose that because it is the base of a right triangle. We can see the angle of the vertex (the shift in position) and now, we do a bit or trigonometry using AU, which we know precisely (as mentioned above)
The distance of a star that shifts by one second, is about 3.2 light years, or one Parallax-Second - PARSEC. Parsecs are what Astronomers use to discuss distance.
Ok, so, how about objects that are farther?
The first big breakthrough was from Sir Edwin Hubble.
First - We knew that certain stars, known as Cepheid variables (because they were first seen in the Constellation Cepheus, the Whale) vary in luminosity. And the period of that variation is proportional to their luminosity. So, they became “standard candles” - we knew how bright they were by timing the period of their oscillations in luminosity.
When we know the absolute brightness, we can tell the distance by how bright it appears. Light spreads out, uniformly, from something like a star, so, the farther away, the more dim it appears.
Next, we knew that stars had emission and absorption spectra. That is, that the light in the star, when separated out, had very clear lines in the spectra that were the result of hydrogen emissions. Now, those emission spectral lines are exact, and they are the result of a quantum effect of electron excitation and emission of that radiation. The energy levels don’t vary so the spectra don’t vary in the absolute.
All stars have hydrogen, so, we can see those lines. Here is where the Doppler effect comes in - if those lines are shifted to a lower frequency (red shifted), then that star is receding from us. If those lines are shifter to a higher frequency (blue shifted) then that star is moving towards us.
Others had predicted that the universe could be expanding (again, the result of Einstein’s work) but Hubble was the first to see it - the farther an object was from us - the more the lines were red shifted. It was a direct correlation, built of of measurement of the red shifted and fixing distance with standard candles.
It became known as Hubble’s Law - and it firmly established that the universe was expanding.
We’ve since developed other techniques, but that’s why Hubble won the Noble prize and why we named that telescope after him.
Before Hubble, we thought that was only one galaxy, and that the fuzzy objects we could see were “nebulae” - but Hubble proved that they were actually distant galaxies, equal in size, or larger, than our own. Minds were blown as the result of measuring distances accurately and correlating them with red shifts.
We’ve since refined the “Hubble Constant” and gotten a better handle on the exact relationship, but the Law, and the relationship, is firmly established.
https://en.wikipedia.org/wiki/Hubble's_law
https://en.wikipedia.org/wiki/Cosmic_distance_ladder#Parallax