Here is the short mathematical answer :
- An ordinal is compact if and only if it is a successor ordinal.
- A cardinal is compact if and only if it is finite.
Here we are assuming the natural "order topology" is used. The result for cardinals follows from the result for ordinals since an infinite cardinal must be a limit ordinal.
If λ is a limit ordinal, then λ+1 is a "one-point compactification" of λ which we might loosely express as :
- λ = [ 0, λ )
- λ+1 = [ 0, λ ]
borrowing our notation from that of intervals in the real line.
The notion of compactification you describe in your question is strongly Euclidean and so does not readily apply to the ordinals in general. However, because it is tempting to view the ordinals as a line in the plane, there are number of nifty pictorial representations of infinite ordinals, drawn in a compact region of the Euclidean plane. Here is one I found via google images of the limit ordinal ωω as a spiral :
Intuitively, this type of Euclidean pseudo compactification would only work for countable ordinals such as ωω, pictured here. One can (almost) imagine extending this picture to a higher countable ordinal by replacing each ordinal in this depiction with a copy of ωω to obtain something like ωω2 perhaps(?) (I'm not sure). Imagining further substitutions beyond that, things become less clear. The first uncountable ordinal ( ω1 ) would be at the "limit of the limit of the limit ..." of this process - if that makes any sense.
This sort of doodling is fun, but unfortunately it does not say anything about the formal, topological view of compactness among the ordinals.
EDIT : When one considers the class of all ordinals, compactification seems wholly inappropriate. Re-reading this post, I think that most of what I've said, after the initial mathematical statements, is waffle. Probably just a shameless excuse to squeeze in a pretty picture. The whole idea of closure is contrary to Cantor's formulation, as suggested by @Conifold 's comment below, even if it is well-defined from a topological point of view.