Are surreal numbers actually well-defined in ZFC?

To expand a bit on my comment:

In set theory one often runs into troubles when defining an equivalence relation whose universe is a class. It is quite possible to have an equivalence class (or all) that have a proper-class many members in it (e.g., all singletons).

The common trick around it is known as Scott's trick, invented by Dana Scott to allow an internal definition of cardinality in models of ZF:

Suppose that $\varphi(x,y)$ describes the equivalence relation (that is, $x$ is equivalent to $y$ if and only if $\varphi(x,y)$ is true), we then define the equivalence class of $x$ as: $$[x]_\varphi = \{y\mid \varphi(x,y)\land\mathrm{rank}(y)\text{ is minimal}\}$$

We make a heavy use of the axiom of regularity, from which follows that every set has a von Neumann rank. We take the collection of those with minimal rank.

Note that $[x]_\varphi$ is not empty. $\varphi(x,x)$ is always true so there is some $y\in[x]_\varphi$ whose rank is at most the rank of $x$, and therefore by the replacement/subset schema $[x]_\varphi$ is indeed a set.

A minor remark about the existence of such $\varphi$ is that in ZF [proper] classes are described by a formula. If $R$ is a proper class which is also an equivalence relation then there is $\varphi$ such that $\varphi(x,y)$ is true if and only if $\langle x,y \rangle\in R$.

Surreal numbers do form a class, but the surreal numbers with "birthday" equal to x are all sets. The birthday concept is basically that $0 = \{|\}$ is "born" on day 0, then $1 = \{0|\}$ and $-1 = \{|0\}$ are born on day 1, and so on to $\omega = \{0, 1, 2,...|\}$ and $1/\omega = \{0|1, 1/2, 1/4, ...\}$ which are born on day $\omega$, and on to infinity. In general the birthday of $x= \{X_l|X_r\}$ is the (ordinal) supremum of the birthdays of all the numbers in $X_l$ and $X_r$.

Restricting yourself to all the surreal numbers with a birthday $x$ or less, where $x$ is some ordinal, you produce a set. On this set you can produce equivalence classes, addition, and multiplication. It can then be proved that these definitions on the set of surreals with birthday $y$ or less, where $y$ is greater than $x$, are extensions of the definition on surreals with birthday $x$ or less. Therefore, the definitions can be extended to as high an ordinal as you want, so intuitively they can be applied to the class produced from the union of each birthday set for every ordinal.

See the Appendix to Part 0 in Conway's On Numbers and Games.