Is it possible to make clouds, in your own backyard?

I know it would be hard, but if you could, how?

Sure... that's essentially what fog machines do, indoors or outdoors. Fog is a stratus cloud.

You can also make a cloud in a bottle:
http://www.instructables.com/id/How-to-make-a-cloud-in-a-bottle/
http://chemistry.about.com/od/chemistrydemonstrations/a/cloudinabottle.htm

Or in your kitchen:
http://www.stevespanglerscience.com/experiment/awesome-dry-ice-experiments

From the inside clouds look very different than from a distance. When you go outside on a foggy day, you're walking into a cloud. What it would take to reproduce that beyond what skyDancer describes would be a way to change the dewpoint temperature and humidity so the air has a concentration of moisture greater than 2 percent. If you want your cloud to hang around for a while after you make it, get a copy of a conversion chart that will tell you where the dew point is at the current temperature and do it when there's no wind.