Hello @Scotty81
I'll echo @AbiDhakal , you should look for the data first, and your local government GIS department is definitely one of the places you should look.
You should also search on ArcGIS Living Atlas. This is an authoritative library of datasets curated by Esri. Next, search on ArcGIS Online. These are datasets shared by Esri users. In all cases, make sure to consult the terms of use to ensure you are allowed to use the data for your purpose.
Finally, search google for keywords like gis, data, hub, retirement, locations, and the name of the area you are interested in.
Here are a few datasets that I found that might be relevant to you: (I don't know what country you are in)
For all of these, click "Open in Map Viewer" or "View Map" to see the data.
Once you have data, yes, ArcGIS software can be used to find all of the locations within a given area.
For example, you can use the Find by Attribtues and Location tool in ArcGIS Online. For this tool, you'll also need data of the boudaries of the areas you are interested in, but these will probably be easier to find than the retirement homes. For example, here is data for USA counties.
If you found a list or table of locations that is not yet mapped, you can geocode that list to create a map.
Another data source within ArcGIS is ArcGIS Business Analyst. If you have more questions about that product, post them on the ArcGIS Business Analyst Questions board.