The Shire was the homeland of the Hobbits, usually they are smaller than Men. They were fairly good archers and quite skilled with stone-throwing. Did the Shire Have any Defense system to protect them from any invasions from their neighbour kingdoms ?
The section entitled "Of the Ordering of the Shire" in the Prologue to Fellowship of the Ring provides your answer.
The Shire had no standing army, but would conduct a muster as required, which would be commanded by the Thain (the similarities with Rohan are observable). In practice this never actually happens:
Aside from that, they did also have the Shirriffs who would also act as a "peace-keeping" force (among other duties) and could even "rustle up a posse" if required to deal with Outsiders:
Per Simon's comment (below), they also had the Dunedain, which took care of a lot of outsiders; however the shire had no control over these. After the events at the end of Return of the King they gain Pippin and Merry.
And that's about it.
Who would invade them ?
Since they reckon more on their discretion than on their ability to fight, mostly because they hate troubles, they don't need any army. If they suddenly need one, they would build up one, if not crushed before, else they won't bother themselves by thinking about it.
You have a human mind, and in The Lord of the Rings, humans build up armies. Every time. But Hobbits are peaceful and troubles are considered as some marginal characteristic. You, as a hobbit, wouldn't want to be seen carrying armor and weapons, even for playing, because elders could think that you are serious, and then you should say good bye to you socail status. No more friends, no more party invitations, neighboors advicing youglings to not come near the guy-looking-for-adventures's garden, ...
It's the same for actual humans about sustainable Development. We know we should reduce our comfort in order to the environment to not weaken around us, but we can't bear this conclusion. We would, in fact, do sacrifices and afforts when we won't have the choice, like the Hobbits would build up an army if they were close to be crushed. But not significantly before that point.