Does a SQL Server job run on the server or the local machine? Does a SQL Server job always run on the server, or will it run in the context of the local machine, similar to a DTS package run from Enterprise Manager on the user's machine?

The job in question calls a DTS package. The package succeeds when run locally; the job fails whether run manually or scheduled.
When a SQL Server agent job is run it is run on the SQL Server machine not on your workstation.

It sounds like you don't have the drivers setup correctly on the SQL Server.

Keep in mind that if you are using a 64bit machine you'll need 64bit drivers and god only knows how DTS will handle 64bit stuff.