To make sense to OP's question, one should define what are the "events" in the probability theory one is considering. In other words, which subsets of the natural numbers are such that finding a number belonging to that set is relevant for our measurements/considerations.
Mathematically speaking, this amounts to defining a σ-algebra in the set of Natural numbers (and every element of the σ-algebra is a subset of numbers representing a possible event; in other words, it is a measurable set of numbers). The natural numbers endowed with a σ-algebra become a measurable space. In a measurable space, one can define a probability distribution, i.e. a function that assigns to each event a given probability. The probability distribution should satisfy, as well as the σ-algebra, some mathematical properties that make the probability theory sensible (in agreement with experience).
Since the natural numbers are a discrete space, the possible σ-algebras on it are rather easy to describe. Take a partition of the naturals in mutually disjoint sets. Then the collection of all the unions of an arbitrary number of elements of the partition is a σ-algebra on the naturals, and every σ-algebra on the naturals is of such form.
The simplest possibility is to take as a disjoint partition of the natural numbers each number by itself. In this way, every set of natural numbers (and in particular every number) is a measurable event. However in this case it is not possible to define a uniform probability distribution on the associated measurable space, since a probability must be σ-additive: given any countable collection of mutually disjoint measurable sets, the probability of their union should equal the sum of the probabilities of each measurable disjoint set. Hence if one takes the disjoint measurable sets to be the natural numbers (possible by our definition of σ-algebra), and the probability of picking any number to be equal and nonzero, the sum of such probabilities is never finite (and thus the probability of picking a natural number would not be one, but infinity!).
On the other hand, if one is only interested in having as events the odd numbers and the even numbers, then one could choose a different σ-algebra. In this case, the easiest choice is to partition the naturals exactly into even and odd numbers. Then the corresponding σ-algebra, i.e. the measurable events would only be the empty set, the set of odd numbers, the set of even numbers, and the set of all natural numbers. In other words, we are saying that for us the meaningful events are that a number is natural (and we want this to have probability one), that a number is even, and that a number is odd (the empty set is the event when we are picking no number at all). For the corresponding measurable space, there is a very simple uniform probability distribution: we set the probability of picking no number to be zero, the probability of the picking an odd number to be one half, and the probability of picking an even number to be one half (from which it follows that the probability of picking a natural number is one).
In this latter probability space (that in my opinion describes fairly accurately the thought experiment described by the OP, since in such experiment the distribution should be uniform, but the only relevant measurable events are being either even or odd), the limiting relative frequency of the event "even number" (to use OP's terminology) is indeed one-half.