I would like the source meaning of the term "yeoman's duty."  I know
it means having worked hard, but I want to know the etymology.  The
derivation.  The literal reference of this metaphor.  Lots of terms
are related to seafaring, like being a loose cannon on the deck etc. 
So who is the yeoman who works so hard, and where does he work, and
where does this phrase come from?

Yeoman is an old British term, more specifically England. Possibly
derives from Middle English, yoman or yeman, a young man.

Several related definitions, such as a gentleman servant to nobility,
or freeholder -- often serving as a foot soldier. This evolved into
the ?Yeoman of the Guard?. In all cases though, as the term migrated
through the centuries, the yeoman was always ranked above labourers
and was held in a position of trust and responsibility. From this
probably arose ?yeoman?s duty?, as being something arduous but not
odious, reliability being implicit.

Best

In my experience, "yeoman's duty" means not hard work, necessarily,
but a duty or assignment faithfully executed.  As the previous
contributor indicated, reliability is implicit.

The Yeoman of the Guard refers to the foreman or, in today's terms, a
NCO (non-commisioned officer) or Petty Officer, i.e. only slightly
elevated above the people he commands.  Foot soldiers were a
notoriously unreliable and lazy lot, often pressed into service
against their will.  The Yeoman would have been one of their own who
proved especially trustworthy and who commanded the obedience of those
beneath him.

Hence, the "Yeoman's Duty" was an assignment given to the Yeoman to be
carried out by him alone, and not entrusted to the rank and file. 
This task may be something that might ordinarily be given  to a common
soldier (not difficult or arduous) but for the importance of it being
carried out with attention and in a timely fashion. Serving Yeoman's
Duty is to faithfully carry out whatever task  is given, even the
menial.

Hope this helps.

BTW,

In today's naval service, at least in the US. Coast Guard, Yeoman is a
clerical rating, or specialty. He (or she) is a Petty Officer whose
specialty is office work.  Aboard ship, the yeoman keeps track of
personnel records, processes the ship's paperwork, and generally acts
as an executive assistant for the ship's Executive Officer.  "Yeoman's
Duty" here consists of managing the 1001 details involved in keeping
the ship from sinking under a mound of paperwork - trivial perhaps, in
comparison to navigating, or making the screws turn, but no less
essential, especially if you want to go on leave or get your next
stripe!