What strenuous exercise can I perform in a cramped space?

Do what you enjoy. Really depends on the size of the space and what you like. There are treadmills, stair climbers and elliptical trainers that can fit in small spaces. Otherwise the p90x program will give great results as long as you are healthy enough to do the program. Also consider a kettle bell work out, gives great cardiovascular and strength results. I would consider a yoga mat and program at least 2x.Week.