Check that you have a handler for.svc files like

<add name="svc-Integrated-4.0" path="*.svc" verb="*" type="System.ServiceModel.Activation.ServiceHttpHandlerFactory, System.ServiceModel.Activation, Version=4.0.0.0, Culture=neutral, PublicKeyToken=31bf3856ad364e35" preCondition="integratedMode,runtimeVersionv4.0" />

Also try turn off any custom 404s

Thanks Daniel.  The config file has a handler for .svc files, identical to the line you posted.  Here is the full node structure where this resides. Other config settings are removed for clarity.  Is that location path of "EPiServer" the place where this node should reside?  The reason I ask is because the URL path used to access that doesn't contain the text "episerver".

<configuration>
  <location path="EPiServer">
    <system.webServer>
      <handlers>
        <add name="svc-Integrated-4.0" path="*.svc" verb="*" type="System.ServiceModel.Activation.ServiceHttpHandlerFactory, System.ServiceModel.Activation, Version=4.0.0.0, Culture=neutral, PublicKeyToken=31bf3856ad364e35" preCondition="integratedMode,runtimeVersionv4.0" />
      </handlers>
    </system.webServer>
  </location>
</configuration>

I turned off custom errors by commenting out the lines shown below in the httpErrors node:

    <httpErrors errorMode="Custom" existingResponse="Replace">
      <!--<remove statusCode="404" subStatusCode="-1" />-->
      <remove statusCode="503" subStatusCode="-1" />
      <remove statusCode="500" subStatusCode="-1" />
      <!--<error statusCode="404" prefixLanguageFilePath="" path="/page-not-found" responseMode="ExecuteURL" />-->
      <error statusCode="503" prefixLanguageFilePath="" path="/maintenance" responseMode="ExecuteURL" />
      <error statusCode="500" prefixLanguageFilePath="" path="/maintenance" responseMode="ExecuteURL" />
    </httpErrors>

Now the http://localhost:28022/IndexingService/IndexingService.svc page displays the standard "404 - File or directory not found" message.

What else can I check or adjust?  Thank you.

You actually need to do a bit more. There might be other *.svc handlers inherited from machine.config or other "parent" config files. Need to remove other handlers before adding your own:

<handlers>
    ....
   <remove name="svc-Integrated" />
   <remove name="svc-ISAPI-2.0" />
   <remove name="svc-ISAPI-2.0-64" />
   <remove name="svc-ISAPI-4.0_32bit" />
   <remove name="svc-ISAPI-4.0_64bit" />
   <remove name="svc-Integrated-4.0" />
    ...
    <add name="svc-Integrated-4.0" path="*.svc" verb="*" type="System.ServiceModel.Activation.ServiceHttpHandlerFactory, System.ServiceModel.Activation, Version=4.0.0.0, Culture=neutral, PublicKeyToken=31bf3856ad364e35" preCondition="integratedMode,runtimeVersionv4.0" />

We had something like that as well.

Fixed it by adding a httperrors section for the indexing service location, setting it to the default, so your custom handling won't kick in.

<location path="IndexingService/IndexingService.svc">
    <system.web>
      <httpRuntime maxQueryStringLength="65536" />
    </system.web>
    <system.webServer>
      <security>
        <requestFiltering>
          <requestLimits maxQueryString="65536" />
        </requestFiltering>
      </security>
      <httpErrors errorMode="DetailedLocalOnly" existingResponse="Auto" />
    </system.webServer>
  </location>

Yes, then you need to do fine-tuning for that location, add http errors override, increase/decrease request limits, etc.

The section for that location is created when adding EPiserver.Search if I remember correctly, so you'd only have to add one line.

The problem with the service was/is, that it throws a 404 when called directly, by design. Which triggered the custom 404 handler, in our case the BVN one.

..and my favorite...remember to upgrade your nuget package to latest version :)

I had an old version of the search package that didn't work with my upgraded Episerver CMS (on stage and production only). Upgrading to latest version of the nuget package helped. So that's something to start with before hacking a lot of configs like I did :)

Thank you very much for clarifying this Jeroen, was pulling my hair out!