It is not possible to pack any two transcendental numbers, but it is possible to pack any numbers which can be represented.
Take two transcendental numbers which have representations. pi can be represented (using Eindhoven notation) as: "4 * ( + : (n ∈ Z) ∧ (n > 0) : ((-1) ^ n) / ((2 * n) - 1))". That's a string, in ASCII and HTML. This string can be represented by an integer. For example, put the string in a computer's memory somewhere, consecutive characters taking consecutive memory locations; the whole block of memory the string occupies can be interepreted as one large positive integer. e can be represented as "( + : (n is an integer) ∧ (n ≥ 0) : 1 / (n!))". And that can be represented as another large positive integer.
Now we have two positive integers representing two transcendental numbers. Two integers can be packed easily. From giving this example, it should be clear that any two numbers which can be represented - in any way - can be packed. One merely needs to encode the alphabet used in the representations, linearise the representation in some reversable way (perhaps adding to the alphabet in order to accomplish this), compose one big integer out of each representation and pack those two integers.
So the only numbers which cannot be packed are those that cannot be represented. (The set of numbers which cannot be represented is a proper subset of the transcendental numbers and is uncountable; the set of numbers which can be represented is countable.) And if you can't represent the number alone, it's hardly a restriction that it cannot be represented along with another number.