Meaning of "sleep" and "shave it through on the grub"

I read in "The White Silent" of Jack London and see this sentence

'Only one day. We can shave it through on the grub, and I might knock over a moose.'

I do not understand meaning of 'we can shave it through on the grub'. Do you explain clearly for me?

And Jack London use 'sleep' in this story, example:

I'm a gone man, Kid. Three or four sleeps at the best. You've got to go on. You must go on! Remember, it's my wife, it's my boy—O God! I hope it's a boy! You can't stay by me—and I charge you, a dying man, to pull on.

So what mean of 'sleep' in this context?

The meaning of shave it (same structure as 'do it' and '[We've] made it!', with crypto-referential 'it') is 'just about manage to ...' (survive / win the match / pass the exam ...).

I've not yet found the expression in a dictionary, but the related 'It was a close shave' is a well-known metaphor. 'X just about shaved it' is used informally in the UK at least to mean '[Side] X were just about the better side [and thus deserved to / their win]':

"The score was irrelevant. Both sides wanted to win the match and Oldham probably just about shaved it...." [The Bolton News_2001]

Another example showing a close victory / achievement / outperformance / overcoming:

He was brilliant; so was she. I've watched the favourite & although Olivia Coleman was very good, I think Close just about shaved it for me, great acting. [tidied] [twitter.com/hashtag/thewife]

'We can shave it through on the grub' is thus 'We can just about manage [until the situation improves] on the food we've got at the moment.'

.......

Sleeps here is a metonym/synecdoche for days, days travelling. It obviously connotes more of the lifestyle being enjoyed/endured than the unmarked term 'days'.

Merriam-Webster gives the broadened senses:

sleep ...

3a : a period spent sleeping

b : night

c : a day's journey

There is a famous canyon in the Southwest (arguably just about) of the United States called 'Ten Sleep Canyon' [TravelWyoming] (though it took the coach my wife and I were on less than a morning to negotiate).