Is it possible to sum an entire column without setting explicit cell boundaries in Excel?

Just leave off the row numbers:

=SUM(A:A)

Note that this expression cannot be placed in column A (that would cause an endless recursive summation which Excel won't allow).

If you instead wanted to sum all of the rows in column A above the current row (say 51) you could do this:

=SUM(A$1:A50)

If you ever move the cell that holds that expression it would change the A50 to whatever is above the current cell while keeping the starting point of the summation (A1) the same.

The same thing can be done if the total is kept above the summed cells in the current row (say row 1):

=SUM(A2:A$51)

In the case of Google Spreadsheets you can do the following:

=SUM(C4:C)

It will sum all rows in C column starting from the 4th row.

In my case the fields I didn't want to include in the range actually did have numbers in them. I was stumped until I figured out that you can just sum the entire column and then simply subtract the fields you don't want:

=(SUM($B:$B)-SUM($B$1:$B$6))

Where the first six rows of column B are numbers but not relevant to the data I want summed, and column B contains an indefinite number of rows of information.