Family Crisis Cash Assistance Act This bill provides funding for the Department of Health and Human Services (HHS) to establish a program within 30 days that provides monthly cash-assistance payments to low- and moderate-income households during economic downturns or disasters. This funding is exempt from sequestration, a process of automatic spending reductions under which budgetary resources are permanently cancelled to enforce specific budget policy goals. During any month the Department of Labor determines that a state or territory is in an economic downturn or the Federal Emergency Management Agency determines that an area is under a major disaster declaration, eligible households in those places receive payments. Payments vary based on household size and income. For households with incomes under 200% of the federal poverty line (FPL), payments range from $2,000 to $4,000. Payments phase out for households with incomes between 200% and 300% of the FPL. In addition, payments may not be garnished or taxed, and jurisdictions may not impose conditions, such as work requirements or asset tests, on the receipt of these payments. The federal government must cover 100% of the payments. States and territories may carry out this program directly by submitting plans that meet certain requirements to HHS for approval. HHS must award planning grants to jurisdictions to develop the plans and cover certain program implementation costs. If a jurisdiction lacks an approved plan or otherwise fails to comply with program requirements, HHS must make the payments on behalf of that jurisdiction.