HOW TO download a "large area" and use it in JOSM ??

http://download.geofabrik.de/osm/ is a good source for planet.osm extracts. Basically you are downloading the osm data for a large region of the planet so you will want to look through the folders and find the smallest area which will contain Chicago, there are state sized regions.

I would recommend picking the bzip version to download since this is an xml format which every openstreetmap tool can use.

Now you have a large area of osm data which is too big, so you can use a tool called osmosis to cut out a square. You can find instructions on the wiki. (You can use the export tab to get the bounding box coordinates).

After that you should have a .osm file covering chicago. I just tested josm with an 80mb extract (Cambridgeshire in the UK, which is probably a bit larger than Chicago), it does slow down a bit but once you zoom in it seems to work reasonably, it used almost 700mb of memory.

First of all, you must be aware that what you will be downloading, or editing in your editor, is raw data, not "a map". You will have a line tagged as "this is a road", another line tagged as "this is a building", but it is not a WYSIWYG editor. After you are done editing, you will have to create a map from your data, a process we call rendering. You will be using a different piece of software for that (Maperitive, Mapnik, Osmarender, Mapgen.pl or something).

Now you didn't say why downloading with JOSM didn't work for you. My suspicion is that you might have tried to download a too large area. Try a city block first, to see if it works in principle. If it is really the size limit that is your problem, you will either have to use the XAPI service (which is reportedly a bit flaky at the moment), or you should download a suitable Planet.osm extract and cut out your area of interest with a suitable program like Osmosis.