So with a lot of extra care about dimension/codimension it seems to be possible to define Chow groups over Spec Z if I understand the above answers correctly.

I may point out that in the book by Elman, Karpenko and Merkurjev "Algebraic and Geometry Theory of Quadratic Forms" (even though the title does not suggest so) they very carefully work out Chow groups, even some version of higher Chow groups. They begin by treating Chow groups *over general excellent schemes* (something you do not have written so explicitly in Fulton), so quite general and only later impose additional assumptions, like equidimensionality, being over a field, and all that. So maybe it is worth having a look at that.

On the other hand, they get a pullback along non-flat morphisms only with the typical more restrictive conditions. This however is crucial for turning Chow **groups** into the Chow **ring**.

So I think the construction of the intersection product [which uses the pullback along the embedding of the diagonal X -> X x X] is another very very critical matter over Z (but according to one of the other answers it can be done, that sounds very interesting).

Last but not least, just maybe another perspective, if one writes down the classical intersection multiplicity of two cycles, that can be done by first multiplying both cycles of complementary codim [so for this we need a ring structure, but let's just assume somebody can give such a structure even over Z, just to find out where we would actually be going], then the product lies in CH^n(X), n being the dimension of our scheme. Now to turn that into the classical intersection multiplicity one could pushforward this cycle along the structural map to the base field,

$X$ --> $Spec$ $(k)$

over a field(!) and $CH{\_0}(Spec k) = \mathbb{Z}$ and we get our intersection number. Voilà.
But if we are proper over Spec Z, we could at best pushforward

$X$ --> $Spec$ **Z**

but $CH{\_0}(Spec(\mathbb{Z}) = 0$, so nothing very interesting seems to result here.
[this argument however only makes sense if the dimension shifts in this Spec Z setup would be carried over analogously, which maybe is also stupid here for the reason that Spec Z is one-dimensional and Spec k zero-dimensional. I am just saying all this, because best and supercool would of course be somebody with a Spec **F**$_1$ having

CH_0(Spec **F**$_1$)= ? (....something, probably rather R than Z)

and that could then be our Spec Z intersection number by giving **F**$_1$ the role of a "virtual base field" and I guess some people say this should link to the Arakelovian one.... but well, that's very speculative]

So I think some people's expectation goes in the direction that the "interesting" way of doing intersection theory over Spec Z needs such a final **F**$_1$-twist.

Note maybe that the classical analgoue would be

P1 <-> Spec Z + (infinite place)

but CH_0(P1) = Z, whereas CH_0(Z) = 0, so we kind of miss something if we just use classical Chow groups over Spec Z. For other questions, classical methods work well even for Z without needing **F**$_1$ or so, for example the étale fundamental groups of both P1 and Spec Z are trivial. But for Chow theory some additional tricks seem to be required.

At least that is my impression. Of course this arithmetic aspect of intersection theory over Spec Z is a kind of different story and it also makes perfect sense to talk about classical Chow groups over Spec Z, so there is certainly nothing *wrong* in having CH_0(Z) = 0, just maybe for some sorts of questions of arithmetical content, this type of Chow theory may not be the right approach.