Is there any way to globally handle error and still able to capture
  exceptions details (message, stack trace) & log request object.
Yes, ASP.NET Web API 2.1 have framework support for global handling of unhandled exceptions, instead of adding try catch block on every method.
It allows use to customize the HTTP response that is sent when an unhandled application exception occurs.
WebApiConfig
public static class WebApiConfig
{
    public static void Register(HttpConfiguration config)
    {
        // ...
        config.Services.Replace(typeof (IExceptionHandler), 
            new GlobalExceptionHandler());
    }
}
GlobalExceptionHandler
public class GlobalExceptionHandler : ExceptionHandler
{
    public override void Handle(ExceptionHandlerContext context)
    {
        var exception = context.Exception;
        var httpException = exception as HttpException;
        if (httpException != null)
        {
            context.Result = new CustomErrorResult(context.Request,
                (HttpStatusCode) httpException.GetHttpCode(), 
                 httpException.Message);
            return;
        }
        // Return HttpStatusCode for other types of exception.
        context.Result = new CustomErrorResult(context.Request, 
            HttpStatusCode.InternalServerError,
            exception.Message);
    }
}
CustomErrorResult
public class CustomErrorResult : IHttpActionResult
{
    private readonly string _errorMessage;
    private readonly HttpRequestMessage _requestMessage;
    private readonly HttpStatusCode _statusCode;
    public CustomErrorResult(HttpRequestMessage requestMessage, 
       HttpStatusCode statusCode, string errorMessage)
    {
        _requestMessage = requestMessage;
        _statusCode = statusCode;
        _errorMessage = errorMessage;
    }
    public Task<HttpResponseMessage> ExecuteAsync(
       CancellationToken cancellationToken)
    {
        return Task.FromResult(_requestMessage.CreateErrorResponse(
            _statusCode, _errorMessage));
    }
}
Credit to ASP.NET Web API 2: Building a REST Service from Start to Finish