Sacramento would cap rent hikes at 5 percent under a measure that qualified Thursday for the 2020 ballot as city residents grapple with soaring housing costs. City officials said Thursday that a petition drive, backed by rent control advocates and labor unions, has gained enough signatures — 44,000 — to qualify for the city ballot in two years. The signatures were independently tallied and analyzed by county voting officials. The “Sacramento Community Stabilization and Fair Rent Charter Amendment” would limit rent increases as well as restrict landlords’ ability to evict renters. It also establishes an elected rental-housing board tasked with monitoring and enforcing rent controls.