The battery door on most Canon DSLRs actuates a tiny little switch that shuts the camera down as soon as you open the door. Therefore, if the door is missing, the camera can't turn on.
The switch plunger is visible on this pic, just to the right of the red circle mark:
https://d3nevzfk7ii3be.cloudfront.net/igi/5REI5Dtt6QfjDrFE.huge
You can do the following:
* Obtain a new door, you can fit it without having to take anything apart, it just snaps in.
* Obtain a BG-E5 Battery Grip, these may be easier to find than a spare battery door. Second hand ones go for as little as $40. It clips to the bottom of the camera and allows you to use an extra battery, or even power the camera from a set of common AA batteries. It also doubles as a portrait grip. It does make your camera a bit larger though.
* In case of emergency (I Need My Camera Right Now!) use an object such as a piece of toothpick or matchstick to keep the switch plunger depressed, and some tape to keep it in place. However, you should remove it before you take out the battery, and put it back only after you insert the battery again.