What is in the space between neurons in a brain?
Not so empty, actually.

The human brain has a mass of ~1.5kg, and volume ~1200cc (a little bigger for men, a little smaller for women). So is heavier than water by a good margin.

While it has Cerebrospinal fluid, that only occupies the subarachnoid space (the space below the skull and above the cortex, contained between two layers: pia matter and arachnoid membrane) and the ventricular system (several spaces inside the brain, remnants of the embryological development of the brain).

Neuron density may vary widely, depending mainly on the particular characteristics of neuron cell types and their interconnections. But besides neurons, there's a lot of infrastructure inside the brain. For example:


Astroglia: They are a type of glial cells which participate in the formation of the blood-brain barrier (supporting the endothelial cells), nourishing of neurons, maintenance of ion and neurotransmitter concentrations, among others. They also keep in place most of the tissue.
Microglia: Small cells with immune (phagocitic) functions inside the brain.
Radial glia: A more specialized precursor cell, that also participates in neuronal migration in the brain.
Oligodendrocites: Cells responsible for the insulation (myelination) of axons.
Neuroepithelial cells: The stem cells in the brain.


Neuroglia, which includes the first four cell types above, accounts for ~90% of tissue in human brain (http://classes.biology.ucsd.edu/bipn140.WI13/documents/Gliamorethanjustbrainglue.pdf). 

Also, when you collect some billions of axons, they can be quite representative in terms of mass and volume. The ratio between white matter and gray matter is close to one for humans, but lower for smaller mammals (http://www.pnas.org/content/97/10/5621.full.pdf).

About the last part of your question, the synaptic cleft (the space between pre-synaptic and post-synaptic neurons) is a salty solution: water with high concentration of sodium and chloride ions (and also calcium ions, neurotransmitters and a lot more). These ion concentrations are fundamental to the generation of action potentials, neural signaling, and the general dynamics of the brain.