The Excel Web Part error on the SharePoint site is: "An error has occurred. Please contact your system administrator"

When looking in the application log on the server I saw:

Event ID: 5226 - Excel Calculation Service
Unable to create or access workbook cache at C:\Windows\TEMP\Excel server\FileCache\SharedServices1-942760090\Ranges. Excel Services is unable to function without a workbook cache.

Googling the error will present you with posts to check that the disk is not full, that the account running the application pool tied to Excel Services has appropriate permissions to TEMP and to try restarting the Excel Services service. Great places to start, but I found my issue was much simpler:

It turns out previous to the error I ran CCleaner on the server, which goes through the system and removes "unneeded" files...which included some of the subfolders in temp. I did have all the structure down to C:\Windows\TEMP\Excel server\FileCache\SharedServices1-942760090\ but missing subfolders. It seems that it's OK to delete all of TEMP (it will simply recreate the cache), but if the subfolders are missing Excel Services gets confused.

So browse to the directory in question under \FileCache\[SSP Path Name]\ you should create 3 subfolders: \Ranges, \Charts, and \SavedWorkbooks.