Finite abelian groups as class groups
The smallest abelian group which is not the class group of an imaginary quadratic field is $(\mathbf{Z}/3 \mathbf{Z})^3$. There are six other groups of order $< 100$ which do not occur in this way, of orders $32$, $27$, $64$, $64$, $81$, and $81$ respectively. The groups $(\mathbf{Z}/3 \mathbf{Z})^2$ and $(\mathbf{Z}/2 \mathbf{Z})^4$ occur as class groups of imaginary quadratic fields exactly once, for $D = -4027$ and $-5460$ respectively. These results follow from the "class number $100$" problem, solved by Mark Watkins.
If you restrict to the $p$-part of the class group, then the answer (for general number fields) is positive. That is, for any abelian $p$-group $A$, there exists a number field $K$ with class group $C$ such that $C \otimes \mathbf{Z}_p = A$. There is even a non-abelian analog of this. Namely, for any finite $p$-group $G$, there exists a number field $K$ such that the maximal Galois $p$-extension $L/K$ unramified everywhere has Galois group $G$. This is a very recent result of Ozaki.
Virtually nothing is known about the question of which abelian groups can be the ideal class group of (the full ring of integers of) some number field. So far as I know, it is a plausible conjecture that all finite abelian groups (up to isomorphism, of course) occur in this way. Conjectures and heuristics in this vein have been made, but unfortunately for me I'm not so familiar with them.
The situation for imaginary quadratic fields is different. Here there is an absolute bound on the size of an integer $k$ such that the class group of an imaginary quadratic field can be isomorphic to $(\mathbb{Z}/2\mathbb{Z})^k$. Conditionally on the Generalized Riemann Hypothesis, the largest such $k$ is $4$. This has do to with idoneal numbers, of which the following paper provides a very fine survey:
http://www.mast.queensu.ca/~kani/papers/idoneal-f.pdf
Actually the truth is slightly stronger: let $H_D$ be the class group of the imaginary quadratic field $\mathbb{Q}(\sqrt{-D})$. Then, as $D$ tends to negative infinity through squarefree numbers, the size of $2H_D$ (the image of multiplication by $2$) tends to infinity. See for instance
http://arxiv.org/PS_cache/arxiv/pdf/0811/0811.0358v2.pdf
for some recent explicit bounds on this.
At least as of 1999, this was still an open question, according to the MathReview of a paper by Marc Perret (On the ideal class group problem for global fields, J. Number Theory 77 (1999), no. 1, pages 27-35; MR1695698 (2000d:11135), review by Bruno Anglès).
Luther Claborn proved in 1966 that every abelian group is the ideal class group of some Dedekind domain (Every abelian group is a class group, Pacific J. Math. 18 (1966), pages 219-222, MR0195889 (33 #4085)).
Gary Cornell (Abhyankar's lemma and the class group. Number theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979), pp. 82–88, Lecture Notes in Math., 751, Springer, Berlin, 1979, MR0564924 (82c:12007)) proved that every finite abelian group is a subgroup of the ideal class group of a cyclotomic extension of $\mathbb{Q}$. There have been some refinements, but I did not find anything in MathSciNet that would suggest the problem has been settled in the interim.