I've used Calculate Areas to get the area of a great-big multipart polygon. All my data and my workspace is in WGS84 so according to the documentation my area should be in square degrees (even though this measure is essentially meaningless). Anyway, the MBR of my data is approx 100 square degrees and yet the area that the tool has given me is approx 8 000 000 000. So what is this number?

I also note the the documentation for Calculate Areas says that it will be depreciated and Calculate Field or Geometry Calculator are better tools. However Calculate Field doesn't appear to give me the option to return an area and I can't find Geometry Calculator - it doesn't appear in the search.

This entry from the help files makes reference to the Calculate Geometry dialog. You can simply use the Field Calculator and calculate geometry either in the units of the file or in the units of the data frame.