To be clear, a 60%-70% overhead rate implies that the university is taking 37-42% of the money (for each dollar spent, another 0.60-0.70 goes to university). In addition, other expenses like tuition and equipment have no overhead taken out.
Equipment is a special case in that "capital" equipment is usually exempt from most or all overhead. Smaller things are often not though, and some things are very difficult to charge to a grant at all.