The first thing to consider is that Russia was a phenomenally large country for an autocratic ruler to govern, much larger than any other European state at that time. Putting Siberia to one side, in 1700, the Tsar's European territories stretched from Kiev on the Polish frontier in the west to the Ural mountains in the east. For perspective, the distance between Kiev and the Ural mountains is greater than that between Kiev and Amsterdam. In 1800, by which point Russia's western territories had expanded greatly, the distance between Warsaw (the westernmost major city in the Russian Empire) and the Urals was greater than the distance between Warsaw and Galway on the west coast of Ireland. Communication was extremely difficult - even after German experts set up a postal system in Russia in the mid-seventeenth century, it still took two weeks for a courier to travel from the capital, Moscow, to Kiev. Even at at the end of the eighteenth century, Russia's road network was still fairly rudimentary and most of the roads weren't even paved. Only the advent of railways and electric telegrams in the 1830s could really change this, and Russia was slow at adopting them, as its failures in the Crimean war of 1854 - 1856 against the technologically advanced, industrial economies of the UK and France showed. Russia was also an incredibly rural state - as late as 1800, only 3% of its population lived in towns and cities - with an incredibly rich and powerful landed aristocracy, some of whose members owned agglomerations of landed estates as big as England. And until well-into the eighteenth century, European Russia was vulnerable to attack from Poles and Swedes to the West and from Turks and Mongols to the south. Governing it was therefore going to be an incredibly stressful affair for even the most talented ruler, and the more successive Tsars tried to develop a more modern, westernised state, the more complicated and fraught things got. This was perhaps most memorably exemplified in the assassination of Tsar Alexander II, whose reforms included emancipating the serfs in 1861, freeing up the press, creating a western style education system, introducing trial by jury and elected local government councils and officials, making the army more meritocratic etc, by a radical group called "the people's will" in 1881, when he was just about to give the Russian Empire its first ever codified constitution. The toll on one's physical and mental health ruling Russia must have taken on the Tsars fortunate enough not to die a violent death should not be understated, and while I can't claim to be an expert on the medical science it would have significantly limited their lifespans, making them vulnerable to strokes and other things. And some of the Tsars' strategies for successful rule were not exactly the most conducive to longevity. For example, Peter the Great (r.1682 - 1725) found that the best way to keep the nobility tightly under his control was to make them all reside in his capital and subject them every night to extremely boozy parties where everyone was supposed to get totally hammered, and if you were caught looking too sober or trying to get an early night the guards would be called over to you and force-feed you brandy or vodka from a bucket with a ladle - his court was called "the all-drunken synod" for a reason. Peter himself died aged 53 of gangrene, no doubt induced by his superhuman capacity for alcohol consumption (he allegedly never had hangovers), and towards the end of his life he showed signs of dementia, reflected in his excessively paranoid persecution of his son Aleksei for several years until he at last had him judicially murdered in 1718.
Yet there is another element to this we should consider. Despite having all the appearances of a hereditary monarchy, the Russian Empire never developed a system of primogeniture for royal succession like France, Spain, Great Britain, Prussia, Austria and most of the other monarchies contemporary to it had. Peter the Great himself came to the throne in 1682 when his half-brother, Tsar Fyodor. Peter did have another, elder half-brother called Ivan, but he was passed over because he was both physically and intellectually disabled and because of the political calculations of Peter's mother's relatives. However, the streltsy (the musketeer regiments of the Russian army) revolted not long afterwards and Peter and Ivan had to be made co-Tsars with Sophia, their sister and half-sister respectively, acting as regent. Peter was only able to properly take charge after Sophia had been elbowed aside in 1689 and Ivan had died in 1696. Peter himself did nothing to rectify the situation. Indeed he threw a spanner in the works when in 1722 he promulgated a new law that allowed the reigning Tsar to choose their successor at will, reserving the right to "remove the one he has appointed in the case of unseemly behaviour, so that his children and descendants should not fall into wicked ways, having this restraint upon them." Since he'd judicially murdered his only son as mentioned earlier, Peter had no obvious candidate to succeed him nor did he use the powers he'd granted himself to appoint a successor. Instead, when Peter the Great died, his closest confidant, Prince Menshikov, took the widowed Tsarina Catherine to the Guards to win their support for her nomination. At that time, the imperial senate was debating the succession and the nobles there seemed to favour Peter's grandson, also called Peter, but the guards put a stop to that and Catherine was proclaimed Peter's successor. It was Peter the Great himself who had given the palace guard regiments the role of king-makers, by placing them permanently in St Petersburg as part royal bodyguard part municipal police force and giving each of their members automatic admission to the court. Indeed, the Guards would be king-makers in every imperial succession from 1725 to 1825 (exactly a century) bar one (which we'll come to). They essentially performed a role similar to the Praetorian Guards of imperial Rome, all except they never went as far as selling the imperial title as the Praetorians did in 193 AD to Didius Julianus before bumping him off 9 weeks later because he didn't have the funds. The most famous of all the Russian rulers to have come to power through a palace coup, against her own husband Peter III (r.1760 - 1762), was Catherine the Great (r.1762 - 1796), and to illustrate how fragile a Tsar's legitimacy could be given how succession worked, she faced a dozen revolts in the first twelve years of her reign led by various pretenders. The most famous of these was Pugachev's rebellion, in which a man claiming to be Peter III (Catherine's deceased husband) led an army of disaffected serfs, Cossacks and Tartar and Bashkir nomads that succeeded in devastating an area the size of the British Isles and torturing, raping and killing 1,527 landlords and their families, 237 clergy and 1,067 government officials until it was suppressed by imperial troops - also indicative of the difficulties of governing such a vast, multi-ethnic empire discussed earlier. Catherine's son and successor, Paul I, was the only Tsar in the period 1725 - 1825 to succeed his predecessor in an orderly manner, and after five years on the throne he was assassinated by the palace guards in 1801 for being too liberal and reformist to the liking of conservative nobles.
Short answer: Romanov dynasty rulers didn’t live long due to a combination of seemingly poor genetics, poor healthcare, a mixture of bad luck and early death you can attribute to the relative instability of their dynasty - compared to, say, the Habsburgs or Hanoverians over in Europe.
Longer answer: While I don’t pretend to be a medical history expert, I do have a background in studying Russian rulers of the Romanov dynasty.
So I’ll focus this answer in two parts - a very brief background of Russia’s Romanov rulers, the undeveloped nature of its data-gathering and medicine, and then a discussion of the lifespans of the tsars (and tsarinas) themselves.
Breaking down the deaths
In total, there were 19 Romanov rulers - that's an average of one ruler every 16 years.
Just two lived into their sixties, Catherine II, who died at 67, and her great-grandson, Alexander II, who was assassinated at the age of 62.
In total, of the 19 Romanov rulers, 6 were assassinated, or died well before old age, after being removed from power. The remaining 13 died of either complications caused by poor health, disease, or, in Catherine II’s case, probably ill-health caused at least in part by old age.
So how did comparable dynasties do in the same time-frame? The answer is: a whole lot better.
Longer living in Europe
A cursory look at British monarchs in the same time frame as the Romanovs (1613-1917) reveals plenty of rulers who lived into their sixties, seventies and eighties, as does a look at rulers of Prussia and later Germany.
So, going back to Russia, aside from the assassinated or deposed ones, were the tsars just unlucky? It’s hard to say.
The tsars were inept administrators. Pre-Soviet Russia was plagued with either incomplete, inaccurate or simply non-existent data - not entirely surprising given the size and scope even the most basic fact-finding mission would have entailed.
For example, the first census of the Russian empire wasn’t carried out until 1897. Even based on sketchy estimates, it wasn’t until 1950 that Russia’s overall life expectancy exceeded the age of 50.
Background to Russian medicine
With its backward, peasant economy, Russia lacked healthcare facilities in general. Russian tsars generally had access to their own expatriate English, French or German doctors - which most of the population wouldn’t have had, Russia lacked the research or background in medical facilities and research.
You could also speculate that the long-held tradition of personal rule over the country - constantly at threat from neighbouring empires such as the Poles, Swedes, Ottomans, Persians and Prussians presented enormous strain on the health of the ruler. This stress would not have likely been felt by other European monarchs, who tended to rule with a far greater level of delegation.
The micro-managing nature of the Romanov monarchs shows best in the last tsar, Nicholas II, who didn’t hire a secretary and chose to answer and reply to even the pettiest of correspondence himself.
British author Sir Arthur Newsholme summed up Russian empire regime medicine in his 1933 book Red Medicine: Socialized Health in Soviet Russia thus:
But although there were distinguished Russian scientists, physicians, and surgeons, the number of practitioners was entirely inadequate.
Another point to bear in mind is Russia’s harsh climate - and the theorised toll it takes on long lifespans.
A closer look at the Romanov rulers
Let's rewind back to 1613, to the the dawn of the Romanov dynasty. After the leaderless, freewheeling time of troubles, which ended with the selection of the first Romanov tsar, Mikhail I at the age of 17, Russia seemed to finally be on a more even keel.
As the father of the dynasty, Mikhail I himself did not seem to be bursting with health, suffering from a progressive leg injury which crippled him at the end of his relatively short life. He died at the age of 49.
Dynasty father Mikhail fathered 10 children. Just four would survive to adulthood - three of them women. The one surviving male, would be Mikhail’s successor Alexis I. Alexis himself died at 46.
Despite his shorter life, Alexis himself would be the most fertile of Romanov tsars, fathering 16 children from two different wives (the first wife died weeks after given birth to the thirteenth child, which itself died at less than a year old.
Numbers game
Of Alexis’ children, 10 would survive to adulthood - yet some of the notable ones that did seemed to all have various health issues. Alexis’ successor, Fyodor III, was intelligent but frail and physically disabled, and died at 20.
Alexis’ other two successors, Ivan V, probably had what we know today as learning disabilities, and various physical ailments, leading one historian to describe Ivan V somewhat unkindly as a ‘halfwit.’
Meanwhile, Peter I, his other son from his second wife, was extremely tall, strong and healthy. He would go on to be known as Peter the Great, and was a lover of all things European, including skilled European medical doctors.
Yet despite Peter I’s vastly better physical condition, the emperor still suffered from a facial tic and almost uncontrollable rages. It’s up for speculation whether his conditions were purely mental or physical.
To summarise, we now have two lines of Romanov blood: the sickly line from Ivan V, and the relatively healthy line from Peter I.
Dynastic struggle
In the upcoming decade of dynastic struggle, Peter I’s dynasty won out - and it was this dynasty that would continue in one form or another (if you don’t count biological intrigues or likely theories about the fatherhood of certain Romanov rulers) through to the very last reigning Romanov emperor, Nicholas II.
The reign of Catherine II, better known as Catherine the Great, marks more than just the longest-living Russian ruler, in well, forever. Despite being very overweight in her old age, as even flattering paintings show, Catherine II would live on to 67.
Catherine II was ethnically German, and had not one drop of Russian blood. Yet of her at least five pregnancies (only one legitimate, and even that one, the future Paul I, rumoured to be the child of one of Catherine’s many lovers at court), only two would survive to adulthood.
Scientific approach
Catherine was a lover of science, and after her husband she deposed had suffered from hideous smallpox scarring, had herself and her son Paul I inoculated against smallpox by a British doctor - a pretty novel procedure at the time.
Catherine’s son and successor, Paul I, would live for 46 years, being killed in a palace coup. His death in 1801 marked the very last of what had been about 80 years of palace coups, many of them dynastic struggles surrounding Peter I and Ivan V’s conflicting bloodlines.
Yet despite Catherine’s robust genetics, her son Paul I’s successors, starting with his own son Alexander I, didn’t seem to fare much better.
Alexander I lived to 47. Nicholas I lived to 58. Alexander II lived to 62, but was assassinated by revolutionaries. Alexander III, who suffered from alcoholism, died of kidney disease at 49. The last emperor, Nicholas II, was deposed and murdered by a Bolshevik firing squad. He was 50.
Soviet successors
Lastly, how did the Romanov’s Soviet successors do, life-expectancy wise? Just like rivalling European monarchs, a whole lot better.
After an initial hiccup when Nicholas II’s de-facto successor, Bolshevik leader Vladimir Lenin, had a mysterious illness and died at 53, Soviet rulers began to live vastly longer lives than any of their Romanov predecessors.
Stalin, despite the stresses of World War II and the considerable toll it played on his health (the aging leader, by then in his sixties, would in his final post-war years spend months at a time on vacation in the warmer south), lived to 74.
And with the exception of Soviet leader Yuri Andropov, who lived to the age of 69, all Russian rulers after Lenin would die in their 70s or beyond.
The ‘curse’ - if you believe there to be one - seemed to be well and truly broken.