Packed me up in his coach:
Pack as in how you would pack something into a suitcase.
and hemmed me in:
Blockaded, such that the protagonist was unable to move freely. E.g.:
The brick path to the door was hemmed in on either side by tall, unkempt boxwood hedges.
with a folding and jingling barrier of steps:
Coaches had folding steps that collapsed away when not in use, these are what is blockading the passenger against the interior of the coach.
The vocabulary is claustrophobic : hemmed, pack, and is being used to convey the sense of claustrophobia the character is experiencing in the coach.
By popular consent and courtesy of Malvolio, here is a photograph depicting a coach of a similar type that Dickens is writing about here:
(Photo source: Gail-Thornton)