Salsa20 expands a 256-bit key and a 64-bit nonce into a 2^70 byte stream - where did the 2^70 come from?
Salsa20 is a PRF working in Counter-mode. That is it generates the keystream as $K_i=F_k(N,\text{Counter}_i)$.
This Counter is a 64-bit value - not the nonce - thus allowing $2^{64}$ invocations of the PRF before repetition. Each PRF invocation yields 64 bytes, so in total $2^{64}\cdot 64=2^{70}$ bytes.

Salsa20 generates the stream in 64-byte (512-bit) blocks. Each block is an independent hash of the key, the nonce, and a 64-bit block number; there is no chaining from one block to the next.

The above is the relevant quote from the paper, about five lines below the quote in the question.