There are several limitations on the size of arthropods, partly related to the square-cube law and partly related to mechanics. All of these limitations have solutions, but the result may not qualify as true arthropods. A general overview may be read in this article.
Known records
According to this article (quote by wikipedia):
The largest arthropod known to have existed is the eurypterid (sea
scorpion) Jaekelopterus, reaching up to 2.5 m (8.2 ft) in body length,
followed by the millipede relative Arthropleura at around 2.1 m (6.9
ft) in length.
Problem: Circulatory systems
According to this article:
Arthropods have an open circulatory system: instead of having arteries
and veins to channel the blood, arthropods possess open sinus where
blood bathes the organs directly. In which ways does this imply a
constrain for a giant insect? While there is no active mechanism that
pumps the blood throughout the body, it would be very difficult for a
giant insect to oxygenate and nourish all its cells due to the gravity
effect.
On the other side, most insects breath passively through their
spiracles, which connect with an internal system of branched conducts
called “trachea”. Thus, they don’t develop any active system to force
air to enter inside their bodies, but it enters passively throughout
these “trachea” and reaches the inner of arthropod’s body to oxygenate
all cells.
Diffusion of gases is effective over small distances but not over
larger ones. So, giant insects would face serious problems to
oxygenate their tissues if they reach big sizes. In addition, current
atmospheric concentration of oxygen (21%) wouldn’t be enough to
oxygenate such a big organism with such a simple breathing mechanism.
It must be said that all these constrains are attenuated in aquatic
ecosystems, where the cuticle’s weight and the diffusion of oxygen
posed no problem for growth. That explains why the world’s biggest
arthropods (and other invertebrates) are mainly located in aquatic
ecosystems.
Solution: Tracheal circulatory system
According to this article:
Maybe the animal has book lungs like a spider, or maybe the spiracles
of an insect have branched inward, becoming an air-filled tracheal
system intertwined with the fluid-filled cardiovascular system of
blood. Each leg has its own “cardio-pulmonary complex” associated with
it, plus a big one in the belly to feed the organs.
Rather than breathing in and out, these animals breath THROUGH, with
air entering the system through spiracles near the head and exiting
near the ail. Air is pumped by action of the muscular blood vessels
that wrap around the tracheal tubes, or by muscular contraction of the
whole abdomen (like a balloon inflating and deflating). Running also
generates more flow-through.
Problem: Molting
According to this article:
In terms of strength, chitin sheaths around legs (what bugs already
have) works fine. Do the math, and you’ll find that a beetle-leg
scaled up to the dimensions of one of my legs ( 100cm long by 20 cm in
diameter) will have a exoskeleton about 0.6cm thick, which about the
same mass and a quarter of the thickness of the bones in my leg.
That’s not bad, especially considering the fun you could have with air
pockets, different materials, and the exact shape of the bone in
question. I’m confident exoskeletal legs will work, at least for an
animal of my size.
The real problem is that an exoskeleton must be shed as the animal
inside it grows. Imagine a lion-sized arthropod molting and going from
armored battle-demon to squishy pink lump. It might not be able to
support the weight of its own organs, let alone run and pursue prey.
Solutions: buoyancy, cocoons, growing, scales
According to this article:
There are ways to solve the problem. Dig a hole and hide in it while
soft. Immerse yourself in supportive water. Build a “mobile cocoon”
out of the old cast-off exoskeleton and silk. Or just have the
skeleton grow with you.
Sea-urchins have exoskeletons too, but theirs are made of hexagonal
plates that can be separated and the interstices filled with an
intermediary material (in this case collagen) that later toughens into
the necessary hardness and rigidity (in this case calcium carbonate).
The bones of our skull (which are exoskeletons, in a way) work the
same way. The difference is that we also have specialized cells
(osteoclasts) than can destroy old bone as well as create it
(osteoblasts), so even once the plates have met to form a skull, the
whole thing can continue to grow as old bone is subtracted from the
inside and added to the outside.
Don’t like that idea? You can break the exoskeleton up into scales,
which lock edge-to-edge like puzzle-pieces, and can be lost and
regrown one-at-a-time like shark teeth without sacrificing structural
integrity (bonus: video-game-boss weak spots!). Muscles that are
anchored to areas with no shell-scale won’t have any leverage and will
be useless until the new shell hardens. The animal will have to
change its behavior, either getting help from its conspecifics or
building a temporary crutch for itself out of found materials (wood?
old scales spun into silk?). Either that, or muscle-anchoring scales
remain un-shed, built into large, dead structures as the animal grows,
like the rattle of a rattle-snake.
Problem: Pin joints
According to this article:
How load-bearing joints (like knees and hips) in large creatures work
is by distributing the load across as large an area as possible, and
by cushioning and lubricating the joint by surrounding it with living
tissue.
Obviously, creatures with exoskeletons can’t surround a joint with
living tissue or they wouldn’t have an exoskeleton. And without that
cushion and lubrication, they’re somewhat limited in the types of
joints they can have. For instance, humans have hinge joints (elbow),
ball-and-socket joints (thigh to pelvis), gliding joints (wrist), and
a few others. Creatures with exoskeletons have, primarily, the pin
joint.
The pin joint, essentially, has a pair of protuberances on one limb of
the joint fit into a pair of depressions on the other limb of the
joint. You can readily see this the next time you’re eating lobster if
you closely examine where the “thumb” of the claw connects to the
“hand” of the claw.
Pin joints are a problem as creatures increase in size because they
place all the force of the joint into a relatively small area. I want
you to stand up, right now, and stand on the balls of your feet with
your heels off the ground. Then, with your back straight, slowly
squat. Feel the pressure in your knees? Imagine that times thirty and
you’ll have an idea what exoskeletal joints would have to resist at
your size.
Solution: endoskeletal joints
According to this article:
any other kind of joint (for example the ball-and-socket joint in your
thumb) would require the hard surface on the inside, which is sort of
the opposite of an exoskeleton.
Conclusion
Giant arthropods cannot exist without high amounts of atmospheric oxygen. Even then their exoskeletons cannot compete with endoskeletons. Overcoming these limitations would require the evolution of a new clade of pseudo-arthropods with a variety of unique strategies.