I have two disc bulgings and bilateral sacroiliitis, can i do exercises like playing football or work at the gym without hurting my self?

Yes.: Yes, if your bulging disc is not symptomatic you can exercise. Supervised exercise by an experienced individual or therapist in this type of rehab is advised. If you would like to play football, and would be injured, you would risk becoming symptomatic with this condition leading to bigger medical consequences.

No impact/contacts.: I would avoid as much as possible those exercise or sports that impart contact, like football or boxing, or impact, such as jogging. Outside of that, everything else should be fine - swimming, cycling (recumbent would be easier on the back), elliptical, spinning, tennis. There's plenty of exercise from which you can choose.