There are many different factors that led to the fall of Sevastopol during the Crimean War, and most of them are the same factors that ultimately led to Russia's defeat. Quite simply Russia was outmatched. Russian weapons were inferior to those of the French and British troops. Russia's military leaders were hampered by the bureaucratic culture of the Russian state and on top of that were just of poor quality. Russian troops also had difficulty moving around the country due to a limited amount of railroads which made reinforcing troops and moving supplies difficult.

While Russia's army itself was big numerically compared to the opposing forces, many of these troops were poorly trained, and were a conscripted force, or came form a background of serfdom. Russian troops were also poorly trained, having been recruited very recently for the war as well as there being a distinct lack of officers to train any new recruits. Serfdom and the so-called 'agrarian crisis' were massive issues at this point in Russian history, and play a key role in Russian military history. Indeed the defeat of the Russian army in the Crimean war was partly down to the low morale of a serf army, and the contradictions they experienced while fighting for a state that controlled them, but also pushed the Russian government to reform the system of Serfdom that would result in the emancipation of Russia's serfs. Russian defeat showed peasants/serfs that they were the ones who took the brunt of military casualties in war, and questions about why they fought for a regime, led by aristocratic military officers who were often inept or corrupt led for calls for the abolition of serfdom. The narrative of the brave peasantry being betrayed by the corruptness and ineptness of the military officers led to huge pressure on the government to reform serfdom, and was one of the major outcomes of Russian defeat.

Where as previously a serf who had served in the Russian army served for life, once the serfs were emancipated the army could now begin to build up a reserve of trained troops. These reserves could be called upon in the even of war but in the meantime were able to return to their villages and be productive members of society and their community. This also reduced the army budget massively, whereas before the army needed to support this permanent massive serf based army, now they were only supporting a much smaller number of active recruits who would eventually be dismissed and a new 'batch' could begin training. Recruits would also be taken from all parts of society and no longer be an obligation limited to the peasantry. These changes were spearheaded by Dmitrii Miliutin who undertook a major process of reform within the Russian army, of which this example is but one.

I hope that answers your question. I think that ultimately the Russian loss at Sevastopol was symptomatic of the larger problems they had both militarily and socially at the time and that these factors all are linked (i.e. social factors affected the abilities/capabilities of the military, and military policies or defeat placed huge social pressures on Russia). I'm not sure if you wanted the long winded explanation of serfdom, but that is one of the more (in my humble opinion) interesting factors and consequences of Russian defeat in the Crimean war.

This might be obvious but the fall of sevastopol could be put squarely on the shoulders of Erich von Manstein as the general who commanded the German forces that took the city/port. Overall the Soviets made a pretty good fight at sevastopol, lasting several months and forcing Manstein to end up fighting a battle of attrition that benefitted the Soviets. The soviet forces at sevastopol were second rate, as many were naval personnel from the Black Sea Fleet. The soviets made good use of the defenses outside of sevastopol, including good defensive terrain and a number of in-ground fortresses that took a great toll on the german attackers. You could argue that sevastopol should have been held due to its strong defences and it being supplied by sea throughout the battle, but the siege took place when Manstein's troops were fresh at the start of the invasion and superior to the soviets they were facing.