I believe that no real objects are actually (exactly) 1 meter long, since for something to be 1.00000000... meters long, we would have to have the ability to measure with infinite precision. Obviously, this can be extended to any units of measurement. Am I wrong?
Answer
You're not wrong. However, there used to be an object exactly $1$ meter long until 1960, because a meter was defined to be the length of a certain platinum-iridium rod at certain conditions. Since then, the meter is defined in terms of interferometry, and now it is specifically the distance traversed by light in vacuum within a certain period of time.
Similarly, the kilogram has a prototype whose mass is $1$ kg by definition. There are some proposals to replace that definition, but it hasn't been done yet.
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