For the purposes of air pressure inside aircraft, they are broken down into two broad categories: Unpressurized - These aircraft are typically piston powered and can't go very high, so do not even attempt to regulate the air pressure inside the fuselage.
Pressurized -
These aircraft do maintain pressurization in the fuselage, but it isn't constant beyond a certain point.
The category determines the answer to your question: Unpressurized Aircraft Unpressurized aircraft are not built air-tight for a couple of reasons:
It is very hard to make something as large as an airplane truly air-tight, and keep it that way.
Over time, seals will start to leak, fittings will loosen, etc.
The fuselage would have to be much stronger to withstand the pressure difference between the inside and the outside.
This would make them heavier and more expensive.
There isn't much need.
These airplanes don't fly high enough to really require a pressurized fuselage and a person can breathe more or less normally even though you aren't at sea level.
(Think of people that live on mountains.)
Pressurized Aircraft
Without going into the details of how an aircraft is pressurized (which isn't really relevant to your question), the air pressure inside the fuselage of a pressurized aircraft is usually maintained at sea level pressure up to a certain point.
However, the fuselage is designed to withstand a particular maximum differential air pressure, so in order to protect it (basically keep it from "popping" when the air pressure outside the airplane drops too low), the pressure is allowed to decrease when needed.
In typical jet airplanes, the pressure inside portions of the airplane (typically the passenger compartment, some of the cargo holds, and maybe an avionics bay) at cruising altitude is maintained at a pressure which corresponds to an outside air pressure at an altitude of about 5,000 - 8,000 feet (referred to as " cab in alt itude "), depending on the cruising altitude and the design of the airplane.
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Many aircraft are not pressurized and therefore not airtight.
The occupants would suffocate, if nothing else, and most unpressurized aircraft do not regularly operate at altitudes requiring pressurization.
In the case of pressurized aircraft , it's actually necessary that the cabin is not airtight .
They are, instead, nearly airtight and can withstand significant pressure differentials.
Cabin pressurization works by bleeding high-pressure air from the engines, which is then run through packs (heat exchangers and air cycle machines ) to cool it.
After conditioning, the air is fed into the cabin.
This is what creates the lower cabin altitude relative to the outside atmosphere.
Because more air is constantly being added to the fuselage, a pressure outflow valve is required and it is constantly dumping air from inside the aircraft .
In normal operations, it's automatically controlled, but can be manually overridden in emergency situations.
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Pressurized aircraft are not airtight, and neither are they constant pressure.
One key facet that would make being airtight impossible is the need to change pressure in the cabin.
To change pressure in the cabin, air is pumped in via the packs (covered in another answer), and air is let out via outflow valves.
The input is fairly constant flow and pressurization is modulated by how open/closed the outflow valve is.
Cabin pressure must change because the airframe has a maximum pressure differential between the cabin and outside (7.8 PSI in the EMB-145) and as you climb the pressure in the cabin must be reduced.
The cabin is generally 8000 ft pressure altitude at during cruise.
Subsequently the cabin pressure is increased to the landing airport during descent.
Accompanying these changes in pressure are temperature changes naturally occurring as result of the pressure change (see: ideal gas law).
You may notice this during descents in regional jets as they tend to get very warm as the cabin is compressed.
To regulate the temperature, you need air conditioning, and so you have packs and air cycle machines to deliver air at the proper temperature to the cabin.
Now that you have to provide air into the cabin, its not worth even thinking about CO2 scrubbers and O2 tanks, because the packs take care of that with fresh air.
That should cover airplanes not being airtight, why they aren't airtight and that the also are not constant pressure.
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The answers so far have talked about the differences between pressurized and unpressurized airplanes, however to answer your question why they're not airtight :
The answer is two-fold, first as egid pointed out, the pressure differential would probably be too great for the hull to withstand.
Second, and in my opinion, more importantly, humans consume oxygen and output carbon dioxide (and water).
Having your ship airtight, would cause everyone inside to slowly but surely suffocate from carbon dioxide poisoning and lack of oxygen as the little oxygen carried within the aircraft slowly gets breathed away.
So you need to keep replacing that air, and the easiest way to do that is to let the old air out, and put new air in.
Edit
As Dan Pichelman points out, people in Subs aren't dropping like flies.
True, they're not, but they are in fact also replacing the air.
Almost.
They're removing the CO2 (using Scrubbers) and adding Oxygen (from oxygen tanks).
It would probably be possible to solve it this way for aircraft as well.
The question remains whether it is feasible weight wise, and if it's easier to just pressurize the aircraft using bleed air and dump the old air overboard.
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In many General Aviation aircraft (small planes that don't go above approximately 12,000 ft.), the air pressure in the fuselage is the same as the outside air.
Most people have no difficulties at altitudes up to 12,000 ft, so there is no need to maintain a pressurized environment.
Specific rules state that unpressurized airplanes have supplemental oxygen available, depending on how high the plane is, for how long, and whether its flying during the day or night.
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I just heard it somewhere but I'm not sure about it.   If it is, then why don't they build airtight fuselages?
Is constant air pressure maintained inside the fuselage?
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For the purposes of air pressure inside aircraft, they are broken down into two broad categories:   Unpressurized, These aircraft are typically piston powered and can't go very high and pressurized. Aircraft can be either pressurized or unpressurized.
Aircraft can be either pressurized or unpressurized.