Can I exclude outliers when calculating mean or standard deviation (small-sample)?

Sometimes the outliers end up being of more interest than the rest of the data. The discovery of penicillin was from studying the outlier.

Can you verify that the outliers are due to technical problems? If you can show that they are impossible values then you have justification for not including them, or you may find something even more interesting when trying to figure out the unusual values.

The general recommendation these days is to not discard outliers without good, external reasons. If you can throw out any values that you do not like, then you can make the remaining data say anything that you want, which is not good science.

If you still do not like the outliers, but cannot show the errors that caused them then you could analyze the data both with and without outliers and show how similar/different the results are. There are also "robust" methods that are less affected by outliers that you could consider using (though you may need to consult with a statistician for those).