(Disclaimer: I'm a support personnel at Overleaf. If this question/answer is inappropariate for TeX.SX, could an administrator delete this and direct similar questions to contact email@example.com?)
Overleaf v1 does have a 60 files limit on Free plans, but Overleaf v2's limit is 2000 files: https://www.overleaf.com/learn/Uploading_a_project#Limitations_on_Uploads
Therefore you can port your project over to Overleaf V2 by clicking on the project's name in your V2 Dashboard.
If you would like to continue using Overleaf V1 with this project for now, you can contact firstname.lastname@example.org with your project's URL for the file number quota to be increased.
If you'd like to download your V1 project, you can click on the "Download as .zip" button at the bottom of the file list panel. You can also git-clone the project if you prefer.
Overleaf v1 has been retired on 8 January 2019. You can still export or download your project files, by clicking on the Overleaf menu icon above the file list panel in the v2 editor, and then on "Download > Source".