Aristarchus "method" mentioned in the answer of @Viktor Blasjo was impractical. It is clear from Aristarchus's book that he never tried to actually measure
anything. His book has to be considered a pure mathematical exercise.
The simplest method of finding distances is measuring parallax (see Wikipedia).
The ancients knew this and they also knew that parallax of the Sun is zero to the degree of the accuracy they could measure. Therefore they could only very crudely estimate this distance from below. The parallax of the Sun is too small to be measured in antiquity, or even much later. So they knew essentially nothing about this distance. The story of measuring Sun's parallax is explained here, for example:
The first meaningful measurement of the distance to the Sun was made by Cassini
in 1672. But even at that time it was impossible to measure Sun's parallax directly: he measured parallax of Mars instead (which is closer, and easier to measure) and used the known RELATIVE distances which were known correctly without the knowledge of absolute distances since Copernicus. So it is sufficient to measure just one distance in the solar system to calculate all the rest. A century later an ingenious method based on observation of transit of Venus was developed and more precise measurement was made as a result.
Remark. I tried myself to follow Aristarchus:-) With much better instruments than those available to him. And I confirm by my own experience the common opinion of serious historians of astronomy that his method is useless in practice.
EDIT. Aristarchus proposed several methods. One of his ideas is to use a very large base for the Sun parallax, namely the distance to the Moon. Consider the triangle SME. To find it up to similarity we need two angles. Angle E is measured directly, and or the angle M, Aristarchus had an ingenious ides: just wait until it equals 90 degrees. At this moment you will see exactly half of the Moon. There are two problems. First, how accurately you can fix the moment when exactly half of the Moon is visible? Second, the angle E you measure is close to 90 degrees, say $90-\epsilon$, where $\epsilon$ is small. It is this $\epsilon$
which you really need and have to measure. But it is much smaller than the accuracy of any instrument Aristarchus could possibly make. (Compute it yourself, using Wikipedia data!)
So any measurement will give some number which can be slightly greater than 90 or slightly smaller, and you can make no conclusion whatsoever from such a measurement. This is why this method is impracticable. His other methods are of the same nature.