To first answer your specific question: much of the populated Western world was surprisingly well mapped by 1900, and even at large scales, mapped features typically are within 10 meters of the positions that could be more exactly revealed in the 1930s by widespread use of aerial photography. But areas outside North America, Europe, British India, and Japan were much less accurate—wildly so for interior Africa, and South America.
Because a version of your question comes up every few months, I have a canned response that—if you'll forgive me—I cut and paste from:
On a sunny day, all you need is a protractor and plumb bob to check the height of the noonday sun and figure out your latitude. At night in the Northern Hemisphere, the North Star gives even more precision. Longitude was much harder, and the invention of accurate clocks was the eventual (most popular) solution. I sometimes explain it to grade-schoolers like this: if you set your clock to London time when you sailed westward, a few weeks later you would notice that the sun was directly overhead when your clock said 6 pm. You're a quarter-day off, and that means you are a quarter of the way around the earth, or 90ºW.
So if you have a goodly number of latitude-longitude observations for various points along a shoreline, you can sketch the parts in between quite accurately—supplemented by taking bearings of the coastline (for instance, noting that a particular mountain or headlands is due east of you when you're at a known point). Coastlines, of course, were visited frequently by folks (sailors) who could move about easily, could see long distances, had the skills to record latitude & longitude—and who had a great interest in knowing exactly where they were. A 19th century atlas will show the Iberian peninsula almost as accurately as a satellite image.
Mapping of interior areas was substantially more difficult. You can use similar readings of latitude-longitude, as Lewis & Clark did to some extent, but it's slow, tedious work. Most early maps of continental interiors were instead based on a form of dead reckoning, with the explorers roughly estimating distances based on number of hours of travel, and recording observations about large bends in river systems based on compass readings or sun position. These sketch maps sometimes proved to have errors of more than a hundred miles when more accurate positions could be recorded.
For large-scale mapping of interior lands, you can also use triangulation to very accurately create a network of known points scattered across an entire nation, and this was well under way in some European nations by the 19th century—but triangulation is really slow and exacting work. From those known points, typically mountain peaks or other things visible from a distance, you can use simple compass bearings to fill in the spaces in between with a little less accuracy but much greater speed. In much of North America, the rectangular land survey system that divided the advancing frontier into townships and sections of farmland quickly filled in a lot of the blank spaces. This was done with a little less accuracy than precise triangulation, but was more comprehensive, because the surveyors recorded on their plats the rivers, forests, swamps, lakes, and roads they encountered while walking the section lines. Here, for instance, is the surveyor's plat of a portion of the Chicago area.
Some previous discussions: