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I'm still a beginner at QGIS and am currently using it to plan trap placement for an upcoming fieldwork session in Central Europe. Traps should be placed along a transect 100m apart. So far when I place them, the Cartesian reading says they are 100m apart and the ellipsoidal reading says they are 65m apart. This is over only a 1km radius so I don't believe the curve of the Earth is having a significant enough impact to cause this difference between the two readings. And when I double check these distances on Google Maps it is clear that the ellipsoidal reading is the accurate one.

Could it be a problem with my CRS?

I chose EPSG:3857 because I'm using a Google satellite base layer which stretches crazily when I used EPSG:4326 but not when I used EPSG:3857.

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    No, it is not possible to get accurate distance from Web Mercator. That is not its purpose. It should not be a surprise that the poles are not infinitely far from the Equator, yet that is what the math of Mercator asserts. So distance in 3857 is meaningless. Deproject and compute spheroidally, or deproject/reproject into an equal-area. – Vince May 6 at 23:29
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Problem of projections and accuracy of measurements: Since EPSG:3857 (and every Mercator projection) heavily distorts lengths (the more so the closer you get to the poles), measurements in this CRS don't make any sense, as @Vince already stated.

Cartesian vs. ellipsoidal measurements: which one is more accurate: This also means that the difference between Cartesian and ellipsoidal method in your measurements are indeed due to the CRS you selected. The correct one (or better: closer to correct one) is indeed the ellipsoidal one as it does not measure distances projected to a flat canvas, but on the ellipsoid that the CRS is based on.

Problems of accuracy with ellipsoidal measurements: But here exactly the problem starts. Not only the projection, but also the underlying ellipsoid can be responsible for a limited accuracy. It is, in fact, a model of Earth's shape, too and thus not 100% accurate. Even closer to Earth's shape is the Geoid - but again, this is a model too. On a next accuracy level, there is also topography to consider etc.

What you should consider for measuring: You have to think about if the ellipsoid that EPSG:3857 is based on (WGS84) is accurate enough for what you intend to do. If you don't care about a few meters, it should be OK. Using a GPS receiver of a normal cell phone for example, you anyhow won't get enough accurate positioning (depending on model you use and the conditions/context where and when you measure up to +/- 15 meters). Accurate enough for a tourist to find London Tower Bridge, but not far from acceptable for an engineer building a road. So it always depends on use case.

In this context, see the Twitter post by @Nyall Dawson, a preview to the next release of QGIS 3.20, announced for release on June 18, 2021:

the war on EPSG:4326 has begun! Now #QGIS gives a friendly descriptive warning when a CRS based on an ensemble datum (like WGS 84) is used.

https://twitter.com/nyalldawson/status/1390118738251317254?s=20

As can be seen on the animated screencast, selecting EPSG:3857 will result in a warning:

Limited accuracy of at best 2 meters

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