How accurate were the maps before the invention of artificial satellites? What was the most accurate map of the Earth before satellites?
The first satellite image of Earth was taken in 1959. At that time, as far as I can tell, aerial photography and stereoplotters were used to produce topographic maps with accuracy that I imagine would have depended mostly on the quality of the aerial photographs, but probably down to a few meters.
For larger maps depicting the Earth, I think the answer depends on how accurately you could determine your position (coordinates) given the technology available. Sextants, invented in 1757, allow you to determine latitude to within 200m, but probably less accurate in practice. Longitude was a bit more difficult historically, but by the end of the 18th century you could determine it to within half a degree (~20-55km) using the lunar distance method and mechanical clocks. So by 1800 and maybe even earlier you could determine your position accurately enough to map the world in a way that would make the shape and size of continents indistinguishable from modern world maps.
Combined with exploration of the Earth by sail ships during the 19th century, and polar exploration in the 1890's and 1900's you probably had world maps that are more or less indistinguishable from modern ones from around 1900 on. Indistinguishable in the sense that the landmasses would have the same shapes and sizes as a modern map of the world with a large scale.
Here are a examples that show how high quality was even before satellites:
The maps were almost as accurate as they are after the launch of satellites. And this has little to do with air photography. And celestial navigation ("sextants" hinted in the previous message) was responsible only for mapping of remote islands.
The main method of making accurate maps was geodesic survey. One begins with laying a base, that is measure distance between two points using a chain. Then this base is used as a side of a triangle whose two angles adjacent to the base are measured with a theodolite (an instrument of much higher precision than a sextant). So one obtains a triangle. Then one uses one side of this triangle as a new base and continues. The points of triangles are marked by special markers (you can still see many of them, especially if you climb the mountains). Almost all Earth area, especially densely inhabited places are covered with these geodesic nets.
A modification of this method also gives heights of places. For example the height of the main mountain peaks was known to meters already in the 19-th century, without any photography and any satellites.
Only for islands not visible from the continent shore (or from other islands that are visible from the continent shore) one had to use astro-navigation. Astro-navigation on land can give you better accuracy than 200 meters, especially with repeated observations.
A refined form of astronomical determination for example was used to detect the drift of the continents. This kind of observation requires large telescopes. We are talking of the speed of displacement measured in centimeters per year here. And all this was measured before the satellites and not using air photography.
National Geographic is generally considered the premier map maker since the inception of the society over a hundred years ago. The National Geographic 1970 World Map probably answers your question of "What was the most accurate map of the Earth before satellites?"
Note that the premise of your question may be off kilter. In most cases it is not the accuracy of a map that is important, but its features. If the town you are looking for is not on the map, then it does not matter how accurate the map is. In general, the legibility and features of a map are of more importance than the accuracy. Accuracy is only important for navigational maps, i.e., nautical and aeronautical charts. A pilot will draw bearing lines on charts to plot a course and in that case, accuracy is important.
Charts are usually on a local scale because piloting generally is a local activity. So, normally there is not much call for a "world chart".
Although I have named National Geography 1970 as the best world map, I should mention that the Soviets were known to have very high quality maps. It is possible they had world charts that were better than NGS maps, however, such maps were considered by them to be military secrets. Some of these maps have been released, so may be obtainable now.
Also, the Japanese are known for making extremely high quality maps. It is possible they have published world maps of better quality than National Geographic, but it is very difficult to obtain Japanese maps from outside of Japan, so I have only seen a few of them. The few Japanese maps I have seen are of unbelievable quality.