Are you comfortable with long division using pencil and paper? My generation was taught how to do this.
There is no problem in principle with carrying out this process using infinitely long decimal expansions. Just do it to ten decimal digits and compute ten (or so) digits of the quotient. Go back and extend all of those calculations to twenty decimal digits. And continue the long division until you have twenty (or so) digits of the quotient.
Repeat ad infinitum. That is one process.
Or, pretty much equivalently, truncate the dividend and divisor to ten digits each. Do an division and write down ten (or so) digits of quotient. Repeat, truncating to twenty digits each this time. Then thirty. And so on. The limit you approach is the true quotient.
If you are a mathematician there is a different approach that can be taken. One way of formally constructing the real numbers is as a set of equivalence classes of Cauchy sequences of rational numbers. [A "Cauchy" sequence is one in which all of the terms tend to end up all being arbitrarily close to one another -- for any positive epsilon (closeness) there is a delta (point in the sequence) beyond which all of the terms are within epsilon of one another].
Two Cauchy sequences are judged to be "equivalent" if one can interleave their terms and wind up with a Cauchy sequence.
Using this construction, pi is the equivalence class of Cauchy sequences that includes the exemplar: ( 3.0, 3.1, 3.14, 3.141, 3.1415, 3.14159, ...). Meanwhile, e is the equivalence class that includes the exemplar: (2.0, 2.7, 2.71, 2.718, 2.7182, 2.71828, ...).
One can define division for real numbers constructed in this manner by taking the limit of the term by term quotients of any two respective exemplars. For instance, pi/e would be the limit of (3.0/2.0, 3.1/2.7, 3.14/2.71, 3.141/2.718, 3.1415/2.7182, 3.14159/2.71828, ...)
A bit of care would need to go into worrying about division by zero and proving that every pair of exemplars yields an equivalent result.
Alternately and perhaps more conveniently one could work to first define multiplication and then define division as the inverse operation. I honestly cannot remember how we did it when I took that class. I think it was this way.
An alternate construction uses
Dedekind cuts. The definition for multiplication is not difficult, but dealing with sign problems makes it somewhat inelegant for my taste. For the product of two positive real numbers, you basically form the lower cut from the set of products of non-negative rational pairs drawn from the lower cuts of the two factors. The upper cut is whatever positive numbers are left over. Division is then defined as the inverse of multiplication in the appropriate sense.