For hyperfinite groups - ultraproducts $G$ of sequences of finite groups $G_n$ - one has the Loeb probability measure, which is analogous to Haar measure in that it is a bi-invariant probability measure. The catch though is that the Loeb measure is not measurable with respect to a Borel sigma algebra (indeed, there is no natural topology to place on a hyperfinite group), but instead on the Loeb sigma algebra generated by applying the Caratheodory construction to the Boolean algebra of internal sets (the ultraproduct of subsets $E_n$ of $G_n$, which have measure equal to the (standard part of the) ultralimit of $|E_n|/|G_n|$).
One way to think of this is that while a hyperfinite group $G$ is not a topological space, it is a sigma-topological space - it has a collection of "open" sets (countable union of internal sets) which obey a weakened version of the topology axioms in which one has closure only under countable unions rather than arbitrary unions. Loeb measure is then the analogue of Haar measure with respect to this sigma-topology. I discuss this a bit in my paper with Bergelson.
One can also generalise from the hyperfinite case to ultraproducts of compact groups, or even locally compact groups if one normalises things properly.