What fills the space between the alveoli in the lungs?
Alveolar tree is like a bunch of grapes. If you put a bunch of grapes into a plastic bag and put this bag into another bag, you can imagine how the lungs are covered.

Alveoli form the surface of the lungs. There is a membrane called visceral (or pulmonary) pleura that covers the alveoli and this membrane continues into another sheet called parietal (or thoracic) pleura that covers the inner side of the thoracic wall. The space between the membranes is called pleural space.

Here you can see how visceral pleura continues into parietal pleura:



Picture 1. Pleural space (source: sdstate.edu) 



Picture 2. from page 1117 of "Cunningham's Text-book of anatomy" (1914) (source: Flickr)