Are modern fighter jets bulletproof and armored? One would not be wrong given part of their operations is being on the receiving end of enemy ammunition shooting at them.
With very select exceptions like the Su-34, and the A-10 if you consider it modern, they are not. Even in these cases only the pilot and a few critical parts are armored. Most of the aircraft's skin offers no meaningful resistance even to small arms.
Extensive armor coverage that could withstand modern air-to-air or surface-to-air weapons simply isn't viable for a fighter for weight reasons. Redundancy is used to resist near misses and light hits instead.
The reasons are different depending on the weapon. MANPADS like the Stinger go for the nearest heat source (engine exhaust). Their capabilities against a fighter are limited to fringe cases (flying too low, caught on landing/takeoff, and a very lucky shot), and armoring the engine nozzle isn't viable anyway.
The lightest effective anti-aircraft weapon, the Sidewinder, uses a continuous rod warhead, a ring of steel expanding at almost 1 km/s. Due to high inertia, it will cut through anything except for the engine, major bulkheads and longerons, and hard elements like the A-10's "titanium bathtub". Aramid lining, which is the only thing a modern fighter could afford weight-wise for full-airframe protection, will be cut.
The heaviest is the S-400 SAM, which intercepts the target at 2 km/s with a 180 kg fragmentation warhead. The resulting hypervelocity impact (3+ km/s) has solids behave like liquids, with impact results determined by mass rather than strength. The low odds of surviving a warhead this size aside, when it's mass that determines these odds, it's always better invested in something functional rather than just armor.
Most mid-size AAM and SAM also fly an intercept course at high velocity and use high-brisance explosives for blast fragmentation. Against resulting high impact velocities, mass is better invested in more redundant hardware, not armor. That is very different from low velocities, common on the ground, where thin layers of high-strength materials can prevent projectile penetration.
Larger fighters are indeed more resistant to damage than smaller lighter ones (see the one-wing F-15). Armoring a few critical elements can help improve survivability, but this armor has to be very localized. The Su-27 family has an armor plate between the engines, so that one wouldn't damage the other if it disintegrates after a hit. Close air support (CAS) aircraft that attack tanks and other ground vehicles carry some protection for the pilot - notoriously vulnerable to punctures, unlike most of the plane - against ground fire.
A combination of these factors is why armor has largely become limited to CAS aircraft: high-strength armor is very weight-effective against the low velocities of land warfare weapons, but much less so against hypervelocity impactors and fragments used in air combat.
See a related question for a few more bits of info on aircraft armor and protection: Why do modern fighter jets use 20mm guns?
Some are. Most notably the A-10 Warthog carries about 1200 LBS of titanium armor up front to protect the cockpit.
A lot of modern fighters have self sealing fuel tanks which is a type of bulletproofing in a sense. It allows the airframes fuel tanks to be hit, but unaffected by bullet fire.
Broadly speaking additional armor adds weight which is generally viewed as a bad thing in aviation.
Aircraft designs also focus on things like redundancy so that they can be hit by multiple rounds of ammunition and have backup systems in place if and when a bullet strike takes out an important system.
Generally speaking, armor is contradictory to flying. Besides, modern fighters fly so high above any small arms fire, at the very least at 10,000 ft, that bullets simply isn't even a very remote concern to fighters. Fighters are much tougher than they were back in WWII, but that's because they must sustain the high G load at high-speed cornering rather than the need to survive gunfire.
However, if we are talking about cannon fire, then it is a very real concern to modern fighters. The US uses 20mm, the Europeans 27mm and the Russians prefer 30mm. Each shot from any of these rapid firing cannons can outright blow a hole in any aircraft, and there is no way any amount of armor against these big guns can be added to fighters without making them too heavy to fly. Fortunately, the unpowered projectiles from these guns can only fly for less than 2 miles before their trajectories start to droop significantly, so generally, fighters rarely get a chance to use these armaments in real combats.
The real threat to modern fighters is air-to-air missiles that they themselves are armed with and surface-to-air missiles. As long as fighters keep away from these threats absolutely nothing can shoot them down. As long as these two kinds of missiles are not a real concern after the enemy air force and SAM batteries are dealt with, fighters can fly with near impunity.
The A-10 is a CAS aircraft, and it is used against cheap targets that aren't worth smart bombs or air-to-surface missiles. CAS is not in vogue anymore, and the USAF keep them because they cost so little, not because they have to fly almost face to face against the enemies. Besides, the armor A-10 is carrying isn't real armor. Real armor stops or bounces back projectiles, the A-10's armor absorbs the projectile without letting it through.