There are some interesting units that really do exist that are seperated by society, or not widely used anymore.
Measure (sun) - I don't remember if it was the Aztecs or Incans, but one of these groups measured the length of the day by the span of the sun.I think there's something like 34, and since they were near the equator, the passing of seasons wasn't as noticable. So one measure was somewhere around 24 minutes.
The Original Roman Calander - The original calander that the romans used was based on a solar day, but a lunar year. Over time, this grew to be a problem as seasons didn't match up with the month (i.e. spring was shifting into summer because the year was 11 days short). Perhaps as something interesting, your 'time' could be inaccurate; pocketwatches often had to be readjusted for being off time because they weren't as accurate based off the releasing of a coiled spring.
The English Mile - This changed with every king for a few centuries until the 1600's. It was because the distance of the foot changed with the length of the king's foot. The Roman mile (another fun one, as Hadrian's Wall is 80 Roman miles, but only 74 SI miles) is shorter than the SI mile by some 400 feet.
Wheels - An English distance measurement of how far a standard wagon wheel rolled in a full rotation. While I don't know the actual distance (that old pi-r-squared thingie) this was a common mode of measurement, as it didn't shift with every monarch. I think it was around 3 yards. I think its origins start with the Romans.
Blocks - We use this measurement still, and it isn't clearly defined at all. Supposedly it is 250 squared feet of city space, but it changes to whenever the next perpendicular street intersects your direction of travel, and has no value in the countryside.
Seasons - Before ancient man realized that the sun didn't go around the earth, some civilizations measured time in Seasons. They may thought that the world was flat and at the center of all this stuff called the universe, but it wasn't hard to count upon Winter and Summer unless you lived near the equator (in which you still have dry and rain season). The word 'year' was substituted with one of the seasons.
Astrological Projection - Since celestial navigation was extremely important to both land and sea travelers (maps hadn't really been invented until neat the 14th century) some things were based upon the movement of the stars, which Venus (the Autumn Star) was one of the Primes. This wasn't accurate at all (Venus has a 255 day year) but it did set a sort of measuring stick.
Lunar Calander - Yes. The calander was once ruled by the moon. 28 day months, 356 day years. Do the math. Put in thirteen months and you got a 364 day year. Just don't mind that you're lagging 5 days every 4 years. (Surprisingly, it was Julius Caeser that figured this out. Thus, the Julian Calender. We now use the Gregorian Calender)