Don't overlook storage that is remotely located from your residence. It may sound odd to drive many miles to your trailer but with a little planning you can make it work. I don't live in Chicago but I suspect there will be times when you want to access your trailer in the winter and perhaps even take it camping to a warmer environment. A one-in, one-out for the season is not very agreeable to those scenarios.
We're in a similar situation in a different urban area. For over 10 years we've stored our trailer and tow vehicle in a suburban (even rural) area near a lake. We found the covered storage near the lake to be much more affordable compared to anything we found in town.
In addition to cost the in town storage areas had other undesirable issues in my mind. Many of the vehicles parked there that were not RVers or boaters so they didn't come in and out once in a while they were in and out every day. I suspect a guy with a Bobcat on a flatbed isn't going to be as cautious as a person with a nice RV or boat.
The suburban/rural RV & boat storage may offer amenities geared toward recreation - a wash area, dump station, air hose, etc.
Plus you're probably not camping near downtown so look for a storage place, even if it's miles from home, that's in a likely direction that you'll be traveling in the trailer.
Lucius and Danielle
1992 29' Excella Classic
1996 GMC Suburban C2500 7.4L
2005 Chevrolet Suburban K2500 8.1L
Got a cooped-up feeling, gotta get out of town, got those Airstream campin' blues...