Can eczema make u sick?

Yes, it could. Eczema is certainly an annoying physical condition but hardly a serious or life threatening one. However, eczema herpeticum, very uncommon complication occuring in individuals with eczema when they are infected with herpes virus, can be serious if not treated promptly.
It depends. It depends on what you mean by 'sick' -- eczema can be very itchy, which is uncomfortable, and it can get infected. Eczema by itself doesn't cause nausea, fever, or other whole-body symptoms, other than the rash and itch.