Show a schematic, and your goal. Is this going to be a product? Is this just a hobby board?
Chances are you'd only need a single-layer board as some have said, I've certainly prototyped this kind of stuff on single-layer PCBs back when I prototyped boards making my own PCBs with the toner transfer method.
Now if you're going to have the PCB manufactured, single-layer almost doesn't make any sense these days, as now only very few companies offer single-layer PCBs, at least for individuals at low quantities (only big companies making millions of products still use single-layer boards for the most part, even sometimes not FR4, for things like power supplies, panel PCBs holding buttons and LEDs, etc.)
With a few exceptions and special deals, 4-layer PCBs unsurprisingly cost approx. twice what 2-layer PCBs of similar class do. And you probably don't need 4-layer here, but again if you can share a schematic and the application, we may have more accurate advice.