I'd like an open source solution for multiple people all editing a
spreadsheet (ideally Excel, which we all have).  I'd like to run the
server that hosts the shared copy on linux -- with an easy
installation that doesn't need root access.

This might help.  It's creating a shared workbook.  Never tried it
myself.  I don't think you need a Linux server to do this.

"A shared workbook allows several people to edit simultaneously."

http://office.microsoft.com/en-us/assistance/HP052622941033.aspx

As suggested by james, you don't need a separate server to do this -
you just need to create a network share on one of the PCs (assuming
file and print sharing are enabled), and those wishing to access the
workbook need the client for MS networks installed.

You could do it using a Linux server by running the Samba server
software, in which case it would appear as though it were a Windows
server. Here's a link describing the process:
http://www.cse.lehigh.edu/~brian/course/sysadmin/notes/CJanneck-Windows.pdf

I hope the brevity of this comment allows a GR to provide a detailed
answer and claim the fee.