Going from Illustrator to GIS can be difficult, as Illustrator uses different formats, doesn't have any capabilities to deal with projections and coordinate systems, and doesn't store attributes.
One route to try is to convert the .ai file to .dwg or .dxf. However, there can be some issues. See this older GeoNet thread:
Convert illustrator vectors to arcgis
There also are some commercial products out there that try to span the gap if one has funds. Here's one of them:
Avenza Releases MAPublisher 9.8 for Adobe Illustrator
Given your specific project, I would suggest a different approach, though, than using Illustrator. Instead, make the pictures digital (scan), then bring them into ArcGIS with the desired projection/coordinate system already set in the Data Frame. Then Georeference the pictures. Finally, create a new feature class and then use Heads-up digitizing to trace the boundaries into the feature class. You can always export back out to Illustrator if the results need to be there for the final product (exporting from GIS to .ai is less problematic than .ai to GIS).
Chris Donohue, GISP
