You can do a remarkable amount of bodyweight exercise with minimal equipment and not a lot of space.

http://www.startbodyweight.com/

Planks, push-ups, and squats of many variations require no equipment at all.

Add dip bars and a chin-up bar/rings and you're basically set.

Variations mean people who've never exercised before can start very small and work their way up to incredibly challenging routines.

For cardio, good ol' walking and running work. And if you try running, don't think you have to immediately start running continuously; interval training is a thing, and is a great way to start small and work your way up.