How close is close enough, with regression?
I think it is a far question, since we should always try to translate any metric result to a useful interpretation.
When talking about regression, we can judge the success of our model by checking whether that MAE or RMSE exceed what we are willing to accept. 
For instance, if we want to predict the temperature for the next days, and our result (in this case in a time series problem type) is a MAE of about 2ºC (after re-scaling our data to the natural units like celsius degrees), is it enough for our goal? Is a mean (absolute) error of 2ºC above or below the real temperature enough for the case being studied? Maybe it is for an approximate model, but maybe it is not a good result if we want to use it in an environment where tempreature could have more impact on whatever being studied...
Another additional chance you have is estimating confidence intervals (see this answer) for your model coefficents and, more precisely, prediction intervals (see this other validated answer) to have an estimate for your prediction together with an interval.