One of the people in our team has left. They were working on a few projects that I need access to, but they were marked private. Can I somehow access them as the administrator?
Hi @Lachlan and welcome to the Forum! 
If you are in a premium account, once you deprovision a member a private project containing all their tasks will be automatically sent to the Admin of your Org; you can learn more in our Guide article: Asana FAQs and common questions | Product Guide • Asana Product Guide.
If this user had Private Projects with unassigned tasks, I would recommend you to reach to our Support Team, they might be able to help you gain access.
I hope this helps @Lachlan! Please let me know if you have any follow-up question!