Why is something not a field if it's a proper class? Why is it a convention to say that, for example, the nimbers and surreal numbers aren't fields because they aren't sets? Is it just pedantry?
 A: Yes, it's pedantry, but of the sort that's necessary for avoiding wrong conclusions.
If you call the surreal numbers a field -- let's call them $\mathbf S$ -- then you would expect to be able to do the things with them that you can do with fields in general. And sometimes those theorems are proved using the assumption that the field is indeed a set, and they don't necessarily hold for the "large" fields you propose.
For example you would expect to be able to form the vector space $\mathbf S^\omega$ of finitely supported sequences of surreal numbers -- so far so good; we can certainly speak about functions with domain $\omega$ whose values are surreal numbers and finitely many nonzero; each of theses is even a set and $\mathbf S^\omega$ becomes a proper class itself.
Now, how about the dual space of $\mathbf S^\omega$? Things begin to creak here, because officially that's supposed to be the collection of all linear functionals on $\mathbf S^\omega$, and since $\mathbf S^\omega$ is proper-class big, each linear functional is a proper class, and then $(\mathbf S^\omega)^*$ is not even a proper class.
Fortunately, the the usual argument will go through that alinear functional of $\mathbf S^{\omega}$ always corresponds to an element of $\mathbf S^{\mathbb N}$, and those things are sets, so $\mathbf S^{\mathbb N}$ is a proper class.
However, now things break down completely. $\mathbf S^\omega$ is certainly infinite-dimensional, and for an infinite-dimensional vector space that is a set, it is easy to prove that $V$ is not isomorphic to $V^*$. But this proof proceeds by counting the elements of the two spaces ...
Edit: Bzzt. No it doesn't. Counting dimensions is enough. Example dies.

Thus we need to keep careful track of which properties of our structures hold in general, and which ones are only for "small" structures whose collections of elements are sets. The "large" structures will generally satisfy fewer properties than the "small" ones will.
In some cases we do indeed do so while developing the theory -- most significantly in category theory and when speaking about models of set theory. In these particular cases, the benefit of being able to speak about "large" structures justify developing a theory of them.
However, for most kinds of structure -- such as fields or semirings -- the need to speak about large instances has not been so great that people usually bother about keeping score about what can be said about them.
A: It's somewhere in between pedantry and mathematically crucial to refuse to put algebraic structures on classes. Let's recall why we have the notion of proper class in the first place: in set theoretic mathematics we want to be able to define new objects by formulas like $\{I\subset R:RI\subset I, I-I\subset I\}$. But the standard paradoxes show that if you start throwing around formulas involving the "powerclass" of subobjects of an arbitrary class, you end up with objects that rarely form a set, and in bad cases are actually contradictory. That's why our logic is based on sets: because to work with arbitrary classes is to work in a system in which not all the axioms characterizing mathematical practice can possibly be valid. 
I chose the above example intentionally: if you're going to study rings, you're going to study their collection of ideals, and its order structure. You're going to want to find maximal ideals to define quotient fields, for instance. But in working with proper classes none of our machinery for dealing with order is valid anymore, and this becomes absurd.
All that's just to make the point that you really ought to have some foundations for your mathematics. But there are foundations in which you can more-or-less talk about things like the rig of cardinal numbers (or Conway's field of surreal numbers!) The simplest one, which is widely used in algebraic geometry and neighboring fields, is to suppose there exists an "inaccessible cardinal" $\kappa$ beyond $\aleph_0$, i.e. one large enough you can't reach it by taking unions of lesser cardinality of collections of subsets of powersets of sets of smaller cardinality. Then all of ordinary mathematics can be done by reinterpreting "set" to mean "set of cardinality less than $\kappa$" and "class" to mean "set". Then the rig of cardinal numbers becomes the rig of cardinals below $\kappa$, but for all algebraic purposes this will look identical to the kind-of-rig you had in mind, with no need for concern about paradoxes.
A: Literally, these things cannot be "fields" because the contemporary "definition" of "field" is that it is a set ... with operations and conditions. That literal disqualification is a bit pedantic, yes, we can admit, for the reason that no immediate indication is given of why it might be a bad thing that the underlying "set" of a field might be "too big".
For many purposes, this issue doesn't matter... e.g., when one only uses finitely-many elements of an extension of a groundfield that has underlying set a literal set.
Probably a prankster could arrange seeming paradoxes from allowing "fields" that were so large that they had no underlying set. Sure...
