There is nothing in the California Building Code that compels public housing (formerly called publicly-funded housing) to provide elevator access.
CBC 11B-233.1 only requires 5% of the total number of dwelling units, including at least one of each type of unit (one-bedroom/one-bath, two-bedroom/one-bath, two-bedroom/two-bath, etc.), to be on an accessible route.
Access Specialist in post #2 has it exactly right: use 11B-206.2.3 exception #4.
I have built many public multifamily housing projects that are 3 story slab-on-grade buildings with no elevator; on every story, all of the units are "flats". The ground floor has an accessible path of travel and contains all of the 5% required 11B mobility accessible units (incl. on of each type), and all the remaining units on the ground floor are adaptable. These 11B mobility units on the ground floor are typically "dispersed" horizontally (not all bunched together in one wing, if possible).
The flats on levels 2 and 3 are only reached via common stairs to the upper level corridors. The units in these upper floors are not required to be mobility accessible or mobility adaptable, because there is no elevator. The CASps we work with typically will spread a few of the 11B sensory/communication units in the upper floors.
What I have described above has been accepted by every plan checker and every CASp that I've worked with.
So, the CBC does not compel an elevator. That said, there may be funding agencies that have their own requirements to provide elevators as a condition of accepting their money. For instance I think TCAC requires elevators for all their multistory senior projects, and they require 50% of senior's dwelling unit to be mobility accessible.
Also, please note that the accessible public housing units must have a path-of-travel to all onsite amenities, so on a non-elevator building you don't put the only laundry room on the second floor, and you don't the only parking in a subterranean garage, unless you plan on building a LOT of ramps. Every feature typically has to occur at ground level.