We have about 2 feet of snow outside right now, and while I don't take the 4 month old out in it, the 8 year old and 18 month old are out in it every possible day. Kids NEED to use their bodies, they need to climb, run, jump, and grow- keeping them cooped up inside and punishing them for following a natural instinct is simply unfair, and setting them up for failure.
Redirect- 'we can't climb on the chair, let's play with X for a few minutes before we go out for a walk to burn off some energy?'
We also have a very small home, but the interior is largely overtaken with kid stuff for now. There is a small slide to climb, a mini trampoline, and a neat fold down target game for throwing small balls and bean bags at.
Honor the fact that a child is not a mini adult- while we may not have the energy to want to climb, run, jump etc- it is a very real and constant need for them. I suspect that if you can find a way to get enough outside playtime, and gross motor use you'll see a rapid improvement in behavior.