A system's wiring constrains its dynamics, yet modelling of neural structures often overlooks the specific networks formed by their neurons. We developed an approach for constructing anatomically realistic networks and reconstructed the GABAergic microcircuit formed by the medium spiny neurons (MSNs) and fast-spiking interneurons (FSIs) of the adult rat striatum. We grew dendrite and axon models for these neurons and extracted probabilities for the presence of these neurites as a function of distance from the soma. From these, we found the probabilities of intersection between the neurites of two neurons given their inter-somatic distance, and used these to construct three-dimensional striatal networks. The MSN dendrite models predicted that half of all dendritic spines are within 100µm of the soma. The constructed networks predict distributions of gap junctions between FSI dendrites, synaptic contacts between MSNs, and synaptic inputs from FSIs to MSNs that are consistent with current estimates. The models predict that to achieve this, FSIs should be at most 1% of the striatal population. They also show that the striatum is sparsely connected: FSI-MSN and MSN-MSN contacts respectively form 7% and 1.7% of all possible connections. The models predict two striking network properties: the dominant GABAergic input to a MSN arises from neurons with somas at the edge of its dendritic field; and FSIs are inter-connected on two different spatial scales: locally by gap junctions and distally by synapses. We show that both properties influence striatal dynamics: the most potent inhibition of a MSN arises from a region of striatum at the edge of its dendritic field; and the combination of local gap junction and distal synaptic networks between FSIs sets a robust input-output regime for the MSN population. Our models thus intimately link striatal micro-anatomy to its dynamics, providing a biologically grounded platform for further study.