Suppose that 2 scientists use different measurement systems in their work. What problems might arise if they shared their data?
The main problem is the misinterpretations when reading each others work. For example, scientist A measures the length of a metal bar in Meters (Say, 5m) and scientist B measures it in Feet (16.4042 feet long), but they don’t write the units they used. They are then asked to create a wooden beam the length of that bar, but they switch their measurements: Scientist A uses B’s data and B uses A’s measurements.
The problem arises when they see there are no units accompanying the numbers. Since they don’t see any units, they assume they are the ones they are familiar with. This means that scientist A will create a 16.4042 meters long beam (Which is 53.8196 feet approximately) while scientist B will make a 5 feet long one (Which is 0.127 meters long, or 12.7cm). The difference between the expected result and what each of them got is huge, which may lead to problems in the construction of whatever the beams were to be used.
A well known case of the problems with using different unit systems is the failure of the Mars Climate Orbiter. The conflict between the data from one software, which used Imperial units, and another one which used Metric units caused the spacecraft to crash on the surface of Mars.
You can read more about it on the Wikipedia article: