In RabbitMQ, queues are quite cheap. They're effectively lightweight Erlang processes, and you can run tens to hundreds of thousands of queues on a single commodity machine (i.e. my laptop). Of course, each will consume a bit of RAM, but unused-recently queues will hibernate, so they'll consume as little memory as possible. In addition, if Rabbit runs low on memory for messages, it will page old messages to disk.
The above only applies to a single machine. RabbitMQ supports a form of lightweight clustering. When you join several Rabbit nodes into a cluster, each can see the queues and exchanges on the other nodes but each runs only its own queues. So, you'll be able to have even more queues! (to the limit of Erlang clusters, which is usually a few hundred nodes) So, a cluster forms a logical broker distributed over several machines; clients connect to it and use it transparently through any of the nodes.
That said, having a single durable queue for each user seems a bit strange: in AMQP, you cannot browse messages while they're on the queue; you may only get/consume messages which takes them off the queue and publish which adds the to the end of the queue. So, you can use AMQP as a message router, but you can't use it as a sort of message database.