@Caroline:
Inadequate rest and excessive stress are both defined as epidemic issues by various health organizations. You are not alone, nor is your situation supposed to be a normal one. You shouldn't "Just push through", that is reserved for the most dire times.
That said, it sounds a lot like your situation is being exacerbated significantly by both those around you and the severe stress you seem to have been dealing with for as long as I can remember you posting here. Like, I'm not sure if anyone's told you, but even fairly low levels of stress can be significantly harmful if they continue for very long, let alone severe issues within your family.
Stress physically harms you, and it's not your fault if you can't keep functioning through it together. You need downtime, everyone needs downtime.
That, said, rest and not doing stuff while many stuffs need doing are not necessarily the same thing. For myself, I find that browsing the internet or playing games does very little to rejuvenate me, really just having the effect of maintaining my current levels of stress/tiredness. My recommendation, in my fully qualified capacity as a random guy on the internet with a few pet theories and some books on tape from an unusually nerdy road trip, would be to get some physical exercise and then sleep.
Common effects of stress on your body
Headache
Muscle tension or pain
Chest pain
Fatigue
Change in sex drive
Stomach upset
Sleep problems
Common effects of stress on your mood
Anxiety
Restlessness
Lack of motivation or focus
Feeling overwhelmed
Irritability or anger
Sadness or depression
Common effects of stress on your behavior
Overeating or undereating
Angry outbursts
Drug or alcohol abuse
Tobacco use
Social withdrawal
Exercising less often