It was growing very difficult to acquire backup parts, training, equipment, and other resources. Even nuclear fuel was getting difficult to acquire consistently resulting in reactors being run at a burnup rate they were never designed for.
A major element of the disaster was the operators wanted to test a safety procedure during shutdown. As they began to shutdown the reactor in the morning another (non nuclear) plant failed under the increased load. Given the critical need for energy (grid running at well over rated capacity due to aging infrastructure) the power distriubtion operator pleaded with Chernobyl plant to delay reactor shutdown which they did.
The problem is they day shift which was well trained in the new safety procedure was replaced by night shift which was unaware of the procedure and a new procedure was tried for the first time with an inexperienced crew in the middle of night.
However the reactor design was the most dangerous aspect.
http://en.wikipedia.org/wiki/Chernobyl_disaster#Causes
* The reactor had a dangerously large positive void coefficient. The void coefficient is a measurement of how the reactor responds to increased steam formation in the water coolant. Most other reactor designs have a negative coefficient, i.e. they attempt to decrease the heat output in the presence of an increase of the vapor phase in the reactor, because if the coolant contains steam bubbles, fewer neutrons are slowed down. Faster neutrons are less likely to split uranium atoms, so the reactor produces less power (a negative feed-back). Chernobyl's RBMK reactor, however, used solid graphite as a neutron moderator to slow down the neutrons, and the water in it, on the contrary, acts like a harmful neutron absorber. Thus neutrons are slowed down even if steam bubbles form in the water. Furthermore, because steam absorbs neutrons much less readily than water, increasing in the intensity of vaporization means that more neutrons are able to split uranium atoms, increasing the reactor's power output. This makes the RBMK design very unstable at low power levels, and prone to suddenly increasing energy production to a dangerous level. This behavior is counter-intuitive, and this property of the reactor was unknown to the crew.
* A more significant flaw was in the design of the control rods that are inserted into the reactor to slow down the reaction. In the RBMK reactor design, the lower part of the control rods was made of graphite and was 1.3 meters shorter than necessary and in the space beneath them were hollow channels filled with water. The upper part of the rod—the truly functional part which absorbs the neutrons and thereby halts the reaction—was made of boron carbide. With this design, when the rods are inserted into the reactor from the uppermost position, initially the graphite parts displace some coolant. This greatly increases the rate of the fission reaction, since graphite (in the RBMK) is a more potent neutron moderator (absorbs far fewer neutrons than the boiling light water). Thus for the first few seconds of control rod activation, reactor power output is increased, rather than reduced as desired. This behavior is counter-intuitive and was not known to the reactor operators.Soviet reactors designs are inherently unstable. They can never be safe on a long enough timescale. It is like a cowboy riding a bull and it requires constant attention and proper response to avoid a disaster.
Western reactors have always operated on a principle of layered passive safety:
1) NEGATIVE VOID COEFFICIENT. As reactor heats up steams if formed which slows the reactor down reducing heat output
2) Gravity lowered control rods. Control rods require both a constant signal from control room and electrical power to remain up. Any failure (operator death, explosion, loss of power) will results in the electromagnets failing and control rods lower by gravity. Relying on two natural forces rather than human intervention.
3) Nuclear poison. Reactor has nuclear poison usually gadolinium nitrate is held back by heat activated seals under extreme pressure. If heat in coolant water rises in bursts the seals flooding reactor with poison. The high neutron cross section absorbs large amounts of neutrons reducing rate of fission.
These 3 systems ensure a reactor will halt even without human response. However that is only part of the safety system. Even scrammed a reactor will put out a "small" percentage of heat, usually 1%-3% in decay heat. Now when you consider a reactor may have 3000MW of heat ouput even 3% is a massive amount of heat which much be gotten rid of or the core will melt. Most of the safety systems in Western reactor designs deal with this problem. Cooling the reactor under adverse conditions (loss of pressure, loss of electrical = pumping power). Once again a layer safety system is used.
Lastly the containment structure can't prevent a core event but it can contain the damage. Cleaning up a core that has melted when the containment structure holds would be a massive and complex operation likely running into hundreds of billions of dollars. Still that pales in comparison to Chernobyl where a lack of containment structure allowed radioactive material to be spread of hundreds of kms.