We began offering the Neo4j Enterprise Edition as a our standard offering in December 2016. For this version, we automatically do daily, weekly, and monthly hot backups (no downtime). We keep the last seven for each time period: 7 daily, 7 weekly, 7 monthly.
For the Enterprise Edition, just run an "Export" from the selected database.
If you are running Community Edition, Neo4j has to shut down to back-up, so we currently do not run automatic backups.
Customers can also use an alpha API we made to trigger exports on your own schedule, effectively lets you automate it. Let us know if you want access to this API.