To a certain extent, yes, we can treat well-orderings of proper class length; but the resulting objects are no longer sets, and treating them is somewhat difficult. And I see no way of getting proper class ordinals.
Let's recall a bit how ordinals are constructed: we began with the notion of a well ordered set, and then found canonical representatives of each well-ordertype (these are the ordinals).
So let's do the same thing here. A class well-ordering is a class $A$ together with a class $R$ of ordered pairs of elements of $A$, such that every nonempty subset of $A$ has a least element according to $R$. (It might see like I should say "every nonempty subclass, but it turns out that using subsets is enough.)
Now this raises two immediate problems:
Talking about specific class well-orderings is easy enough; but how should I write something like "Every class well-ordering is ..."? That would involve quantifying over classes, which can't be done in ZFC.
Moreover, there's no obvious way to develop canonical representatives! So even though class well-orderings make sense, there's no obvious notion of "class ordinal".
That said, we can treat class well-orderings to a certain extent; see e.g. this mathoverflow question or this other mathoverflow question. But the point is that things - even really basic things we take for granted - which are true about well-ordered sets and ordinals, can be false for class well-orders.