Other than the modern base ten system and the base twelve system which things like feet are measured in what other mathematical bases have older societies used?

Civilizations in Mesoptamia used a "Base 60" system, though elements of Base 10 creeped into it. This is because of the 59 numbers, which go into one of the places in the system, were built from a 'unit' symbol and a 'ten' symbol. You can see that in this image where there is a symbol for 10, and a symbol for 1, which they are all created from.

So in Mesoptamian mathematics, numbers are expressed using sexagesimal place-value notation, which operates analogously to our decimal notation, with each number represented as a Cuneiform sequence of digits, and each of these combination of digits represents a value between 1 and 59 inclusive, and every digit is associated with a power of 60, which decreases from one to the next digit.(Mesopotamian mathematics didn't include a 0 until the Achaemenid era onwards, which meant the Babylonian sexagesimal notation is relative, and the the power of 60 corresponding to each digit is not indicated, for example 1 and 60 have the same notation, and it must be worked out from the context, which we do not usually have)

If you would like an example of how to read sexagesimal notation, here one is. Lets put it in context for our own at first, take the number 5678, in a decimal (base 10 system) this number is

(5 x 10^3) + (6 x 10^2) + (7 x 10) + 8.

This is easy and you wouldn't need to do that, however now lets look at one in a sexagesimal system take the number (in a notation seperated by commas) 1,57,46,40 this represents the sexagesimal number

(1 × 60^3) + (57 × 60^2) + (46 × 60) + 40

which, in decimal notation is 424000. (This example is taken from directly from source 1).

Of course you can still see examples of this, such as the way we keep the time , or measuring angles in degrees.

Many Central American societies, such as the Aztecs, Mayans, or the Olmecs used a base 20 system. You can see the Mayan numerals here. The Mayans also had a zero as a place-holder numeral system, for use in its Long Calendar.

A theory is that this number system arose because of people counting on there fingers and there toes. The "columns" were 1s, 5s, 20s, and then the same multiples repeated, again in pairs: i.e. 20s and 100s, 400s and 2000s, and so on. (in the same way we have a 1, 10, 100 etc "column")

Just look at the Mayan Calendar. Calendrical dates have 5 'places' as in 13.0.0.0.0 (or Dec. 21, 2012). Going from left to right these indicate units of time (B'ak'tun, K'tun, Tun, Uinal , Kin).

Kin = 1 day,
Uinal = 20 Kin, (20 days)
Tun = 18 Uinal (or 360 days ~ 1 year),
K'tun = 20 Tun (~20 years),
B'ak'tun = 20 K'tun (~ 400 years).

The Long Count is not pure base-20, however, since the second digit from the right rolls over to zero when it reaches 18.

There are many other societies that use similar or different number systems, at least partially, those are just the main ones. I can give more information if you'd like, but I'm just at work right now (Hence why a lot of my answer has been copied from a previous one of mine).

Sources