Learn English – What does “week” mean in place names

see me through in this sense means sufficient or sufficient to provide.

Often you'll find ...enough to see me through ...

Forty cans of beer is enough to see me through to the end of the week

meaning I have enough beer to last until the end of the week.

To use it without the extra bit at the end the period or event that I have to 'see through' should be known already.

Have you got enough beer for the match tonight?

Yeah, ten cans will see me through.

There is an example here and some explanation that may help. It's not the easiest term to find on the internet, don't mix it up with see [right] through me.

Apostrophes are used to indicate dropped letters in a word. We are mostly familiar with them in contractions like don't and isn't but they can be used in other places too.

Once upon a time they were used to indicate shortening of words by omitting their initial part. It was once common to write phone (a shortening of telephone) as 'phone. Another common use was 'bus (short for omnibus). Both, of course, can now be used as words in their own right.

In your passage the author is attempting to convey that 'skins and 'scaphe are short forms of longer words. 'skin is probably short for diveskin. 'scaphe might well be short for bathyscaphe, but without knowing the work I can't say for sure.