Depends upon hosting provider and the package/account you have with them and what additional services they offer.
If on a shared host ... multiple customers on one server ... more than likely out of luck. Provider will more than likely recommend to you an 'upgrade' to a VPS. Have to check with provider about VPS's (managed/un-managed space, memory, etc. ... see * below).
IF provider has 'attached storage' then one could attach the storage of size you choose (commercial not free) and then use the attached storage for filedir of moodledata. All the other directories in moodledata would still reside on the same server where your code is hosted. For that one would need ssh access to server - an not at all sure C-Panel offers any icon for such a set up.
OR ... shop around for a 'better' hosting place that is more 'moodle friendly'. Many advertise Moodle installs as one-clicks ... those should be avoided ... kinda like jumping out of the pan into the fire there.
* Do take a full site backup ... code tar ball, database dump, and if you can tar ball of at least the filedir directory in moodledata to get an idea of your space requirements if you shop for new hosting. Realize that's sorta a catch22 but ....
'spirit of sharing', Ken