Base 3:
Okay, so, let's say you want to put your house number (from your address) on the door of the house. So you go to the hardware store and buy digits with adhesive on one side and sparkly-reflective stuff on the other side, to attach to the door.
Now, let's say you'd like to be able to represent any number up to ten thousand (or any other arbitrary limit) without another trip to the hardware store. How many digits do you have to buy? Answer: If using base 10, you need four of each of the ten possible digits, roughly, total of 40. (That's not the exact answer because we don't put leading zeros in house numbers.) If using base 2, lg(10000)≈13.28, so you need 14 of each of the two possible digits, total of 28. What about base 3? Log₃(10000)≈8.38, so you need nine each of three digits, total of 27. Just slightly better than base 2.
Have you learned differential calculus? You can write a formula for the number of digits you have to buy as a function of the base, and then you can differentiate the formula and set that to zero to find the base with the minimum possible cost, and it turns out to be e. Since e is between 2 and 3, but closer to 3, that's the minimum practical (integer) base.
Continue what? Using binary? Of course, when it comes to building actual computer hardware, there are many reasons for binary besides transistors, e.g., arithmetic function circuits are built out of Boolean function circuits, and true/false is a binary set.
But in lambda calculus, remember, we're optimizing for proving theorems. And if you want to prove theorems about rational numbers, what you want is a pair of integers, each of which is represented as a Church numeral.
Real numbers are a lot trickier than I think you realize. Streams, yes, but streams of what? Digits, left to right? If so, the promise in your stream has to know how to compute the next digit. That's a little tricky. No, a lot tricky, except in easy cases such as square roots of integers.
But the fundamental problem is much deeper than the tactical difficulty. Even if your computer has an infinite amount of memory, so you can compute any positive integer as a Church numeral, it still can't compute any real number, not even close, because the real numbers are non-denumerable. See this thread for that discussion. If you think it through, you'll see that all of the numbers representable in any fixed-width format such as IEEE floating point are rational; you can't even represent a single irrational number in floating point.
But if you do want to represent a real number as a stream, you'll be better off making a stream of approximations rather than a stream of digits. Two streams, actually, one approaching the value from below and one approaching it from above. That set of two streams is called a Dedekind cut and it's one of the standard ways to define a real number.
One reason that approximate values are better than digits is that digits are a property of the base as well as of the number, whereas the approximate values are numbers, not tied to a particular representation. So maybe two consecutive approximations differ only in the fifth digit down from where the previous two differed, but you don't know that until you compute one or two moves ahead of the digit you're going to emit next. Or another pair of consecutive approximations differ in the same digit as the previous pair. (For example, your sequence of approximations goes ... 3.10, 3.1275, 3.1348, 3.14000, etc.)
But, again, the main reason is that each term in a sequence of approximations has a clear relationship to the desired value (namely, being near it), whereas it's hard to come up with an explanatory theory about why the 200th digit of pi should be one thing rather than another.