How do I avoid the "#DIV/0!" error in Google docs spreadsheet?

Wrap your formula with IFERROR.

=IFERROR(yourformula)

You can use an IF statement to check the referenced cell(s) and return one result for zero or blank, and otherwise return your formula result.

A simple example:

=IF(B1=0;"";A1/B1)

This would return an empty string if the divisor B1 is blank or zero; otherwise it returns the result of dividing A1 by B1.

In your case of running an average, you could check to see whether or not your data set has a value:

=IF(SUM(K23:M23)=0;"";AVERAGE(K23:M23))

If there is nothing entered, or only zeros, it returns an empty string; if one or more values are present, you get the average.