Does Dropbox keep files on its own servers?

This is direct quotation from a website that is useful for your question:

"The company uses more than uses more than 10,000 physical servers to manage user content, along with Amazon Web Services. User metadata is stored in the company's data centers, while the actual files reside on Amazon's S3 storage service.  Dropbox also uses Amazon EC2 instances to help the data centers "talk" to its cloud storage."

You can also read the FAQ section of official Dropbox website here: https://help.dropbox.com/accounts-billing/security/physical-location-data-storage