>How does TSM handle files that are larger then the tape size? ...

It spans to the next available volume - assuming that there is another one
available.  (The Segment numbers in 'Query CONtent' output reflects the
spanning.)

...
>IBM 3494 Library with 3590E tape drives
>
>I am using IBM "K" tapes that are 80GB uncompress and somewhere around
>120-140Gb compressed. ...

Actually, 40 GB uncompressed, with that tape and drive combination.

>... ANR8945W Scratch volume mount failed .
>... ANR1405W Scratch volume mount request denied - no scratch volume
>             available.
...
>I added a few scratch tapes and it just seems to go through the same
>process...

The messages indicate that you really don't have any scratch tapes.
(You can use the mtlib command to physically verify having such volumes.)
Exactly how did you add scratch tapes?  Ideally, you would have done
'LABEl LIBVolume ... CHECKIN=SCRatch', for the volumes to be labeled with
your drives and their microcode, and associated with the library definition
and ending up with that library's Category Code for 3590 scratch volumes,
which is one more than the SCRATCHCATegory value given in your 'DEFine LIBRary'.
(Server option ENABLE3590LIBRARY must also be in play.)  If all of the above
looks okay, verify that your Devclass specification makes sense for the drives.

  Richard Sims       http://people.bu.edu/rbs