Family and Medical Insurance Leave Act or the FAMILY Act This bill entitles every employee to a family and medical leave insurance (FMLI) monthly benefit payment of two-thirds of the employee's regular pay, limited to a maximum of $4,000, for not more than 60 days of qualified caregiving. The bill establishes the Office of Paid Family and Medical Leave within the Social Security Administration to administer the FMLI program. An FMLI benefit payment must be coordinated with any periodic benefits received under a state or local temporary disability insurance or family leave program. The bill imposes a tax on employers, employees, and self-employed individuals to fund FMLI benefits. It also establishes the Federal Family and Medical Leave Insurance Trust Fund to hold tax revenues.