Dozens of elderly residents will lose their homes when a 124-year-old San Francisco nursing facility closes this spring, officials announced Wednesday. St. Anne’s Home on Lake Street will close April 8 due to staffing shortages and rising operational costs, according to the Little Sisters of the Poor, who run the facility. St. Anne’s Home has provided care to the elderly poor, regardless of race, religion, or national origin, since 1901, the group said.