Yes, I think it can be done.
First, let me direct you to this question and its answers. One point that's well made in Ram's answer is - you are assuming everyone will survive. Or even most people will survive. Your world, with three year summers and two year winters, is milder than the assumptions in that question (which mentioned six years)... but a lot of people are still going to die, especially the ill or sickly, oldest and youngest, the poor (including beggars, the crippled, those without sufficient safety nets), and some percentage of serfs or peasants, with the expectations that these people will, over time, breed enough to cover the loss. It would be rarer to lose the richer, the better connected, those with valuable training, nobles and royalty (and their favorites). Their deaths would still occur, but the danger to them would be somewhat offset by money, that enabler which could let them afford the very best in preservatives, techniques, and an abundance of many rare things that would increase their chances of life - like honey, which by itself lasts decades and preserves beautifully (Crystallized fruit might contain nutrients for years). Or the best in whatever foods do keep nutrients, cheeses, meats, dried fruits, and fresh whatever can be foraged or hunted or force-grown in the winter. This happened in real medival times, it can happen in yours, too - and with the continuation of the upper/controlling levels of society, the society itself will continue despite the loss of those considered expendable.
With a scale of three-to-two for summers and winters, - um, wait, how does fall/spring fit into this? A few extra years for the full four-season's cycle, or carving out time between the two extremes? Nevermind, with the season cycle weighted to have more growing season than dead season, it's likely that the amount of food produced during the whole 5-year-year is sufficient for a 5-year-year... if they produced in each month of summer, as much food as our medieval society produced in each month of summer, and stored for the winter the same percentage of food our medieval folk did, and lost the same percentage to spoilage, they would find they had relatively more food (by three-to-two) as our medieval folk had for the winter. Of course, they would probably eat the extra in summer instead of saving it and either be stronger to survive the winters or feed more people in the summers and lose more people in the winter, but the point remains, enough food can be produced.
The problem is preserving it for a scenario where the growing doesn't happen in smaller regular intervals, but instead the results of a single much longer growing season must last for a much longer timeframe during the winter season. They would need a lot more storage space, minimum 5-8 times as much (8x for going from 3 months to 24 months of stored provisions, or 5x for storing for a five-year-year). At that amount of space, they may prefer to build stone or earth warehouses, thickly insulated with sod, instead of the more usual dug cellars which might destabilize houses at 8x. This would be less efficient during the summer, since it would be less insulated and therefore less temperature-stable, but the cold weather during the winter would help keep everything preserved.
So, lets look at foods and their preservation. First, there are foods that are already preserved for two years or longer, that in fact are considered better for longer aging - cheeses, meads and wines, long fermented or cured foods (which today are gourmet or luxuries), could be preserved for the two years required. Of course, these will also be expensive - but the upper class could afford them, which is kind of the point.
Then, there are more common foods that can be made to keep a long time. The methods of the time include drying, liberal use of salt to brine or cure, smoking, and pickling with vinegar, verjuice, or lemon. The use of nitrates, used in preserving meats, was also known to extend shelf life - in medieval times, this would most likely be saltpeter, which formed naturally in stables and caves and could also be produced with some effort anywhere manure could be gathered and aged. Saltpeter was most famously known for its use in black powder, which was sometimes used to preserve meats.
One common example is salt cured meat - barrels of salted pork or beef could easily last a year and possibly be good for multiple years (one reason it was so common on medieval ships), and salted cod lasts for years, salt cured and smoked hams or procuitto would take a year to cure, and could last months after that without refrigeration, and some very dry jerkies, fish, or sausages will keep almost indefinitely.
Then we get to drying. Dried beans will keep almost indefinitely, although after a year or two they will lose their stored moisture, and need longer to soak and cook - though they do not lose their nutrition. Dried grains will vary, some sources suggest 6 months to a year for grains(although that is stay fresh for, not go bad after), others suggest 8-10 years for soft grains and 20-25 for hard grains if kept dry, cool, and out of sunlight. Probably the truth lies somewhere in between - grains that are not dried spoil quicker, and grains may lose nutrition and flavor after that six months to a year (but still be edible and have nutrition, just past its "best-by" date), and those dried and kept in favorable conditions may last much longer than expected.
And, after a grain has been ground into flour, it could be further preserved by making hardtack - dry, twice baked crackers (also known as ships' biscuits) which lasted at least a year in the very poor conditions found on ships, but which might last and be eaten fifty or a hundred years later if kept in better conditions (without killing or starving the eaters).
Fruits can be stored by drying, either in the sun or an oven. Such will last at minimum 3-6 months at peak condition (at room temperature), and will still be edible possibly for years afterwards with less flavor and nutrition, and can have its shelf life extended by storing at colder temperatures. The same is true of dried vegetables (sun-dried or oven dried), 6-12 months for best flavor and nutrition. Of course, storing at colder temperatures, as would occur during the winter time, will extend shelf life considerably - at least doubling the time it can be stored without additional loss of nutrition.
Pemmican was made from meat that was shaved thin and dried thoroughly, and mixed with rendered tallow for storage. Confits, or potted meat was salted, cooked and stored in its own fats. Rendered tallow or suet was shelf-stable, could last for years, and could be used to block air or water from reaching already preserved food (which spoiled them faster). Pickling vegetables or meats often done with a brine of water and salt, and vinegar or verjuice for better preservation, and herbs and spices added for flavor - such pickles would last for months, and the nutritive value was as high or better in fermented pickles as it was in fresh vegetables, although often lacking in vitamin C.
Additionally, it was known foods lasted longer in cold storage - medieval folk without access to refrigerators could still use this knowledge by utilizing mountain caves or packing food in cellars with snow and ice during the winter - which would use the cool temperatures of the underground rooms and the packed mass of snow and ice to keep the foods cold long after the snows melted... or even mining snow and ice from high altitudes to keep these foods cold even longer. In your two-year winter world, that means much of the food stored during the winter would essentially be frozen for much of the time, which can double the shelf life of many dried or otherwise preserved foods.
And, finally, not all food production would halt in the winter. Some animals would be kept long into the winter season, fed on stored grass or hay, acorns, or perhaps supplemented with stale or ruined grain, and slaughtered periodically (except for breeding animals) to reduce demands on winter feed. Pigs were a choice animal to keep for the poor, since they are not fussy and could supplement scraps with forage even in the winter. Hunting was also an option, with game animals reserved for nobility and hares and rabbits and such the prey of commoners. A certain amount of fishing should also be possible in the winter, ocean fishing fleets or individual fishing in rivers or ponds as the temperature allowed. There may have been some winter foraging, (pine tea was used by iron age vikings, it was a winter source of vitamin C). There may have been some gardening, using glass gardens or heat produced from decomposing manure (I admit the time is wrong, but the idea is there), the products of which would be mostly for the rich, but there would be sources of fresh nutrition to balance out the loss of nutrients and flavor from the longer-preserved foods.
So how would the society have to change? A lot, a lot more planning. People would have to regularly store food during the long summers, and have the discipline to store even when there would be plenty of food after the stored food needs to be eaten (using the stored food from earlier harvests throughout the summer, so all of the later harvests can be fresh-stored for the winter). And they might produce more in the longer growing season than simply the summers back-to-back could account for, since they could stagger harvests (and share the labor of plowing and harvesting when it doesn't have to happen at the same time), and have time to forage, and have fewer crops vulnerable to, or growing seasons shortened by late or early frosts.
Possibly the planning might take the form of a heavier tithe for the lords, who also might distribute some extra back during the winter, at least to keep enough people for summer farming, when it's harder for people to manage that extra planning for themselves. There would also probably be more time in winter to pick up artisan skills and handicrafts (needed to afford fresher nutritious foods), to tell stories, to figure out ways to survive and improve their lives - things which did happen in winter, but like the growing season of summer, magnified by the stretch of time it covers.
Also, preserved food will be losing nutrition and flavor as the winter drags on, and so many people will suffer malnutrition and starvation by the time winter's over even if they have enough unspoiled food to survive - they will have to recover during spring and summer. Better preservation methods or winter foraging or growing methods would be a priority - although it is possible that many foods would be more durable, since seeds (grains and nuts) would have to survive the winters in good enough condition to sprout, and the animals would adapt to surviving the longer season cycle as well, making the whole world relatively the same as ours since it had adapted to all the season lengths.
