User blog comment:Deedlit11/Googology in Magic: the Gathering/@comment-5150073-20130412144609/@comment-1605058-20130416145849

State space has two meanings - in abstract machines theory it is space of all states given machine can be in. More generally, it is space of all possible states whole system (machine+storage) can be in. In terms of Turing machines, state space is set of all configurations of machine tape, state and position. Ikosarakt's argument wasn't fully valid because, as state space if often infinite, each of them is finitely representable.

Storage space is some space which can be accessed and amended by machine. It doesn't have to be infinite, e.g. Minsky register machines consist of finite number of counters. Each counter is storage cell. Storage space in Turing machine is simply its tape.