The number of machines needed to run Burp Suite Enterprise Edition very much depends on the scale of your intended usage.
You can run all of the components on a single machine, including the bundled database. This is suitable for evaluation purposes and for many production use cases. On a machine with substantial resources, this setup should be able to comfortably support between 5 and 10 agents. The diagram below shows a single-machine deployment:
Alternatively, you can run agents on several different machines, and you can use your own external database for storage. This lets you scale the number of concurrent scans to be indefinitely large, and utilize any existing database infrastructure that you have. The diagram below shows a multiple-machine deployment, with an external database and separate agent machines:
Note that the Enterprise server and the web server are always deployed on a single machine.