They're not packing anything. They're the ones packed into that space.
The speaker is complaining that they might not be able to do their work because they'll be packed into that little space. They need space to do the job but they won't have enough space, because they will be packed into that little space.
ODO definition of "pack":
2.1 (often as adjective packed) (Of a large number of people) crowd into and fill (a place):
"a packed Merseyside pub"
Let me rephrase your example:
Yes, I suppose we'll have to get down to work soon, though I don't know how we'll be able to if we're gonna be packed into that little space.