Family and Medical Insurance Leave Act or the FAMILY Act This bill establishes the Office of Paid Family and Medical Leave within the Social Security Administration (SSA). The bill entitles every individual to a family and medical leave insurance (FMLI) benefit payment for each month beginning on the first day of the month in which the individual meets the criteria specified below and ending 365 days later (benefit period), not to exceed 60 qualified caregiving days per period. An individual qualifies for such a benefit payment if such individual: is insured for disability insurance benefits under the Social Security Act at the time an application is filed; has earned income from employment during the 12 months before filing it; and was engaged in qualified caregiving (any activity for which the individual would be entitled to leave under the Family and Medical Leave Act of 1993), or anticipates being so engaged, during the 90-day period before the application is filed or within 30 days after. The bill prescribes a formula for determination of an individual's monthly benefit, as well as for the maximum and minimum amounts. An FMLI benefit payment shall be coordinated with any periodic benefits received under a state or local temporary disability insurance or family leave program. The bill prescribes criteria that make an individual ineligible for an FMLI benefit payment and specifies prohibited acts by an employer and penalties for violations. The bill establishes the Federal Family and Medical Leave Insurance Trust Fund. FMLI benefit payments shall be made only from this fund. No amounts from the Social Security Trust Funds or appropriated to the SSA to administer Social Security programs may be used for FMLI benefits or administration. The bill amends the Internal Revenue Code to impose a tax on every individual and employer, all self-employment income, and every railroad employee, employee representative, or railroad employer to finance the Federal Family and Medical Leave Insurance Trust Fund for FMLI benefits.