If I have understood what you are looking for, SUMPRODUCT will allow you to do this:
If your conditions are in row 1, i.e. $A$1, $B$1, etc.
=SUMPRODUCT(($A$3:$A$10=$A$1)*($B$3:$B$10=$B$1)*($ C$3:$C$10=$C$1)*($D$3:$D$10=$D$1)*($E$3:$E$10=$E$1 ),$F$3:$F$10)
Note that the rows have to be identical (in this case, rows 3 through 10). And you cannot use the generic A:A as a range.
This finds all items that meet criteria 1 in cell A1, criteria 2 in cell B1, etc., and then this will sum all values in F3:F10.
Adjust your ranges as needed.