Most gas water heaters do not require AC power.
They need gas, water lines in and out, and a flue.
The flue will be your biggest problem.

Sorry for the dumb question, but is that a flue similar to a fireplace flue? We currently have venting above the water heater, but none below (which, I'm told is not up to current code and will have to be changed when we replace anyway).

Most residential water heaters use a 3 inch flue pipe.
The flue serves the same purpose as a fireplace flue, to carry the products of combustion outside the living space.

If a gas appliance is installed in a restricted area (and just about ANY closet IS a restricted area) it requires vents to supply the combustion air.
Two vents are required, one near the floor and one near the ceiling.
These vents can be as simple as opening in the walls to a larger area in the house or actual 'pipe' that leads to the exterior of the house.
These supply the fresh air to the unit so it can burn the gas correctly minimizing the production of carbon monoxide.

If the intakes are not large enough the heater can have a hard time developing a draft in the flue pipe and spill combustion gases into the living space, including carbon monoxide.

Creating a new flue can costs many hundreds of dollars depending on where and how it has to be run to get outside the house.
Most of the time they must be run all the way to the roof.
There are options that can exit lower through a wall, or even be power vented.
If you get a power vented water heater it will NOT operate when the electricity is not available, unlike a plain jane gas water heater that needs no electric power.

Thanks! I think we should be okay with creating a second vent below because of the unfinished crawl space/footings underneath the house (no cement foundation). This was very helpful. If I want to add a recirculating pump, will the gas water heater continue to operate even if the pump won't work because of an outage (we're in So. Cal. so we have electrical outages for various reasons -- something to consider).

I think you should at least consider an outside mount gas tankless water heater. Your application sounds ideal for it.

You have high inlet water temps due to your location and the outside mount also would work well in your environment.

The outside mount eliminates your flue and solves the problem of where to put the heater.

Hear is a link, also note the box at the bottom of the page that the unit mounts into.

I think the issue of not having hot water when there is a power outage is overblown. Taking a shower isn't normally the first thing I need to do when the power goes out.

Here is a link that might be useful: Outdoor tankless

I really wanted a tankless water heater. MY GC told me it would be cost prohibited because he would have to change the gas line to a large pipe.

Changing the gas service to the house would probably make it unreasonable.

However, I would do the math to see if it was really necessary to change the service.

Many of these older houses were built with gas service to serve inefficient appliances. With the switch to better furnaces and such, many of them can handle the extra demands of a tankless without service upgrades.