If there were two groups:

One group who saw a yardstick before estimating its length
Another group who did not see a yardstick before estimating its length

Would this be a matched pairs design (a person from both groups estimates at the same time, then the next pair, etc.)? Or does a matched pairs design only apply to things like age and gender?

This doesn't seem to fit a matched pairs design with the information given. Usually pairs are matched on some factor (like age, gender, height) within a group of subjects.