Yes!  We offer the ability for you to use your own storage for a flat $10/month.  We provide the ability for you to paste in links to your files, which we then mask and track.

Once you sign up for the $10/mo or higher plan, you'll see the following setting under Settings >> General Settings

Switch it to the option above and you're all set!

Important note:

While we do mask your links in the front-end, the server request will go directly to your file (we aren't able to generate temporary links on 3rd party servers as we do with internal hosting). For enhanced security, Amazon users can augment their AWS bucket policy to only allow referrals from your FetchApp domain. 

Info on AWS bucket policies here.

If you do require your file urls to expire, we provide this feature with our hosting. Be sure to check out our hosting plans for pricing.