Assuming a 64 bit instance (that is more memory hungry) and that the average paste size is 512 bytes, every 2 million documents require 1GB (just tested with redis-benchmark).
This means that if you have a paste every minute, in order to use 1 GB of memory you need to wait 4 years.
Many problems that at a first look appear to be hard to treat with an in-memory DB at a closer look appear quite addressable.
That said this is the kind of problem where the working set is very small compared to the whole set of documents stored, and where documents are rarely written and often accessed in a read only fashion, so a *SQL DB would work super well for that use case.
I Redis server could form a very good addition in order to perform real time stats about the accesses to the document: number of times the document was read, a sorted set for latest created documents, top documents for this month by page views, and so forth.