Two mutually equivalent definitions of an Ulam number are given in Section Weak inaccessibility of real-valued measurable cardinals.
Added in response to a comment by the OP: First here, the first of the two referenced definitions can be restated, more compactly, as follows: a cardinal $\alpha$ is an Ulam number iff for every set $X$ of cardinality $\le\alpha$ there is no nonzero finite nonatomic outer measure $\mu$ over $X$ such that all subsets of $X$ are $\mu$-measurable.
As for how to regard the condition "that the density character (i.e. the minimum cardinality of a dense subset) of every metric space is an Ulam number", one may note the following:
In the same section of the referenced Wikipedia article, we find:
a cardinal that is not an Ulam number is weakly inaccessible
Further, in the first paragraph of Section Models and consistency, we see
ZF together with "there exists a weakly inaccessible cardinal" implies that ZFC is consistent. Therefore, [weakly] inaccessible cardinals are a type of large cardinal.
Further yet, in the first paragraph of article Large cardinal, we find:
The proposition that such cardinals exist cannot be proved in the most common axiomatization of set theory, namely ZFC
Thus, the existence of a metric space whose density character is not an Ulam number cannot be proved in ZFC. So, the condition that
the density character (i.e. the minimum cardinality of a dense subset) of every metric space is an Ulam number
might be considered a mild restriction -- say, in the sense that the density character of any metric space that one may encounter "in practice" will be an Ulam number.