The condition to not have access to a ruler or a clock is actually not very restrictive.
Use 2 electrons and use a very precise forcemeter.
Measure F and use the formula to deduce r, the distance between the 2 electrons.
Put one electron at each end of the pencil to answer the riddle.
Now if your question is can we determine a one meter distance without making any kind of measurement, I don't have the answer.