Net Core 3.1 Return a Task with a result that includes details of the exception?

The comments in the original example shows what you should actually be doing.

you would need to create a model to store the desired information.

Something like

/// <summary>
/// Represents the result of an ADO.Net operation.
/// </summary>
public class AdoNetResult {
    private List<Exception> _errors = new List<Exception>();

    public bool Succeeded { get; protected set; }
    public IEnumerable<Exception> Errors => _errors;

    public static AdoNetResult Success { get; } = new AdoNetResult { Succeeded = true };
    public static AdoNetResult Failed(params Exception[] errors) {
        var result = new AdoNetResult { Succeeded = false };
        if (errors != null) {
            result._errors.AddRange(errors);
        }
        return result;
    }
}

The interface however would need to be refactored

public interface IAdoNetRepository {
    Task<AdoNetResult> FillAlarmsDataTable();
}

and implementation

public async Task<AdoNetResult> FillAlarmsDataTable() {
    string connectionString = _config.GetConnectionString("Data:DefaultConnection:ConnectionString");

    try {
        SqlConnection connection = new SqlConnection(connectionString);
        string cmdText1 = "SELECT * FROM [dbo].[Alarms] ORDER BY Name";

        // Create a new data adapter based on the specified query.
        dataAdapter = new SqlDataAdapter(cmdText1, connection);

        // Populate a new data table and bind it to the BindingSource.
        alarmDataTable = new DataTable {
            Locale = CultureInfo.InvariantCulture
        };
        await Task.Run(() => dataAdapter.Fill(alarmDataTable));

        // Return what ?
        return AdoNetResult.Success; 
    } catch (Exception ex) {
        // Return the task with details of the exception
        return AdoNetResult.Failed(ex);
    }
}

With that done the original code should work as intended

public async Task<IActionResult> OnPostUpdateAlarmDataTableAsync() {

    var result = await _adoNetRepository.FillAlarmsDataTable();

    if (!result.Succeeded) {
        // log the exception returned from the task that failed!
        //result.Errors
    }

    return new JsonResult(result);
}

An alternative would be to let FillAlarmsDataTable throw any exceptions that it may encounter

public Task FillAlarmsDataTable() {
    string connectionString = _config.GetConnectionString("Data:DefaultConnection:ConnectionString");

    SqlConnection connection = new SqlConnection(connectionString);
    string cmdText1 = "SELECT * FROM [dbo].[Alarms] ORDER BY Name";

    // Create a new data adapter based on the specified query.
    dataAdapter = new SqlDataAdapter(cmdText1, connection);

    // Populate a new data table and bind it to the BindingSource.
    alarmDataTable = new DataTable {
        Locale = CultureInfo.InvariantCulture
    };
    return Task.Run(() => dataAdapter.Fill(alarmDataTable));
}

capture (catch) the exception where it is being used and respond accordingly

public async Task<IActionResult> OnPostUpdateAlarmDataTableAsync() {
    try {
        await _adoNetRepository.FillAlarmsDataTable();
        return Ok(); //200 OK
    catch(Exception ex) {
        //...log the exception returned from the task that failed!

        return new ExceptionResult(ex, includeErrorDetail:true);
    }
}