The answer to the question is that it is independent of ZFC, if one is speaking of outer measure.
The right context for the question and its answer is the very active research area known as Cardinal Characteristics of the Continuum. The point of this subject is to investigate exactly how the dichotomy between countable and continuum plays out in situations when CH fails. For example, in this area, researchers define a number of cardinal invariants:
The bounding number b is the size of the smallest unbounded family of functions from ω to ω. There is no function that bounds every member of the family.
The dominating number d is the size of the smallest dominating family of functions ω to ω. Every function is dominated by a member of the family.
The additivity number for measure is the smallest number of measure zero sets whose union is not measure zero.
The covering number for measure is the smallest number of measure zero sets whose union is all of R.
The uniformity number for measure is the size of the smallest non-measure zero set.
The cofinality number for measure is the smallest size of a family of measure zero sets, such that every measure zero set is contained in one of them.
Remarkably, none of these numbers is provably equal to any other. In addition, there are models of set theory separating each of them both from ω1 and from the continuum. In the case of the uniformity number, this is the answer that Jonas Meyer has pointed to above.
One can define similar numbers using the ideal of meager sets in place of the ideal of measure zero sets, and the relationships between all these cardinal characteristics are precisely expressed by Cichon's diagram. In particular, no two of them are provably equal, and there are models of set theory exhibiting wide varieties of possible relationships.
There are dozens of other cardinal characteristics, whose relationships are the focus of intense study by set theorists working in this area. The main tool for separating these cardinal characteristics is the method of forcing and especially iterated forcing.