In a best out of 3 match, one in a forest, one on a ship at sea, and
one on the moon, which side would win in a five on five, fully armed,
all out battle between pirates and ninjas?

This will be close.  

Forest: Ninjas will use their athletic and stealth abilities with
impunity. They will handily win this one.

Ship At Sea: The water is a pirate's natural territory. They might
find difficulty with Ninjas jumping around but their sea weapons and
swords will get the job done.

Moon: This is where it will get tricky with little gravity. Pirates
will have their swords, but Ninjas again will use atleticism. Both can
float around but Ninjas will be more at home with this ability.

Ninjas win 2-1!

It depends on whether it is a film by notorious producer Joseph Lai.
If that is the case, we'll never know who the winner is because Mr.
Lai usually makes Ninja films by taking half a dozen cheap old Hong
Kong movies from all genres, cuts and combines them as he likes,
shoots a few ultra-cheap new sequences with ridiculous backyard-Ninjas
and sells the result as a new, exciting Ninja movie. But since all
characters are actually acting in different film and only new dubbing
makes us believe (ha!) they were interacting, none of them is really
victorious.

I wouldn't be so sure that ninjas will will the forest battle. 
Consider this: Shiver me timbers!

Pirates 2
Ninjas  1

Pirates, hands down.   <-----------biased opinion, I'm Norwegian.

The answer certainly depends on weather by "fully armed" you do mean
fully armed. In this case it would not be much of a competition as
both parties would have ranged weapons, but while pirates are very
likely to have single-shot flintlock pistols that do not suffice in
killing a very shifty athletic character as a ninja AND do not
recharge fast, ninjas would employ a mindbogling range of ninja stars,
crossbows, throwing claws, throwing knifes, caltrops, even small small
needles winth triangle-shaped fins that could be spit into the
opponent's eyes, all of which are retrievable, easily rechargeable or
plentifull.
My view is that in a historically accurate confrontantion, regardless
of the setting, ninja warriors have the best chances of winning and
here are my justifications:
 Ninja warriors, although mainly trained as spies, were superb
fighters, widely considered to be the first "special forces" in
history.
 Pirates, although portrayed as fierce, sharp swordsmen were mainly
thieved and relied on surprise and fearfull appearance to reach their
goals rather than on fighting skills.
 While the ninja's training and engaging in action were permanent, the
pirates would have to cross blades only now and then. For most of
their time, they sailed.
Pirates would be accustomed to charging into battle, playing it
offensively, but ninja were precisely trained to fight in deffensive
conditions.
 The arsenal of a pirate would consist in a one-shot pistol, one
rapier or dagger and, not always, a long sword. The arsenal of a ninja
would be much more consistent in terms of weaponry, plus their
expertise in martial arts, which objectively, could be enough.
   So to conclude, come rain, shine, forest, ship, moon, I say Ninja 3- Pirates 0!

Forest: Ninja

Sea: Ninja or Pirate tossup

Moon: Ninja

I'm for ninjas since they're more skilled, mysterious and disciplined
than any pirate could be. And besides, Microsoft is after the pirates.

There's absolutely NO context. The Pirates win hands down, even Google says so:

Pirates VS Ninjas
http://www.googlefight.com/index.php?lang=en_GB&word1=Pirates&word2=Ninjas

Pirates at Sea VS Ninjas at Sea
http://www.googlefight.com/index.php?lang=en_GB&word1=Pirates+at+Sea&word2=Ninjas+at+Sea

Pirates on the Moon VS Ninjas on the Moon
http://www.googlefight.com/index.php?lang=en_GB&word1=Pirates+on+the+Moon&word2=Ninjas+on+the+Moon

Pirates in the Forest VS Ninjas in the Forest
http://www.googlefight.com/index.php?lang=en_GB&word1=Pirates+in+the+Forest&word2=Ninjas+in+the+Forest

LOL, that was supposed to read:

There is absolutely NO CONTEST.

Sorry 'bout that. I was excited.

In a best out of 3 match, one in a forest, one on a ship at sea, and
one on the moon, which side would win in a five on five, fully armed,
all out battle between pirates and ninjas?

FOREST - NINJA, those crafty SOB's hide in trees and jump 12 feet
high, pirates will be too busy staring at forest wenches and drinking
alcoholic beverages.

SHIP - NINJA, ninja still kill the pirates on a ship, Pirates will be
too busy drinking beer and staring at mermaids.

MOON - Ninja, ninja still kill the pirates on the moon, the pirates
will be puking their brains out after all the beer they drank! as for
the ninjas, they will be stealing the pirates BOOTY!!!!!!!

Ninjas WIN! 2-1 (assuming all ninjas and pirates have oxygen tanks)
Ninjas Win! 2-0 (assuming all ninjas and pirates have no oxygen tanks)

Pirates win.

When can I expect the $2.00?

There is no *AIR* on the moon.

Pirates dead, Ninjas dead. Tie game.