Here's a link to the video, for those who haven't seen it
Sort of. I can't speak to every single part of the video, but here are my thoughts on most of it. General note, though - it's hard to talk about ideal body types when all of the women are wearing bathing suits instead of the foundation garments that gave them the ideal figure. ETA: It's also hard to define what the standards of beauty are - what men find sexy in artistic nudes does not always match up fully to what standards women are holding themselves to in everyday life.
Ancient Greece, 500-300 BCE: It could be just the poses, but she seems a lot more curvaceous than eg Aphrodite of Knidos. I think they're exaggerating the plumpness a decent amount.
Italian Renaissance, 1400-1700: First of all, that's a HUGE range of time to sum up with one body (and the 17th century is Baroque anyway). Most portraits of the earlier part of the era show thinner figures with a high, smallish bust - even when a nude's painted with a fuller figure, she still tends to have small, high breasts. Later on in the period, you see really elongated figures in art, which may be more about the art itself than a standard of beauty. I think they're exaggerating plumpness a lot, possibly because they're taking Rubens as their stereotype despite him not being part of the Italian Renaissance.
Victorian England, 1837-1901: Another too-long range. For the most part, the fashionable Victorian figure was short and well-rounded (especially in the arms), but not full-figured. Around the 1870s, height became desirable, and a low waistline. (Also, her hairstyle is Edwardian.)
1920s: The text isn't far off (flatchested, bobbed hair, no waistline), but technically it wasn't a boyish figure that was wanted but a preteen girl's.
Golden Age of Hollywood, 1930s-1950s: They are very wrong here. Again, this is too large a range for one ideal figure. 1930-1945 saw an ideal figure with broad shoulders and no hips, very slender; after this, shoulders came in and hips came out, but movie stars were still quite slender. The video exaggerates the fashionable curvaciousness. They're going for a Marilyn vibe, but Marilyn's body was much smaller than the popular conception of her would make it seem.
1960s: Pretty accurate.
It's mainly anecdotal. What they're going for, I think, is to show that beauty standards have changed and to promote body positivity ... but while many eras have had a softer and more rounded standard of beauty than our own, they're presenting an inaccurate picture.
They do cite references for a few of the things stated in their accompanying article for the video, but overall, its based on anecdotal evidence, or an oversimplification of their sources. They've also exaggerated a lot of the body types and their time frames are really too broad to build a proper consensus, as /u/chocolatepot said.
Speaking on a just a few of their chosen eras, a lot of their evidence is based on the periods' art, which is problematic. While art is definitely a good starting point, artistic movements often followed their own trends, removed from what was going on culturally.
A good example is mannerism, a movement that emerged during the mid to late 1500s, as part of the High Renaissance and remained until the Baroque era (or a tiny part of 'The Italian Renaissance', if you're Buzzfeed). With an emphasis being placed on expression over realism, the art's subjects were often distorted to suit the mood of the painting, with little consideration for current beauty standards or even the subjects being used. One of Buzzfeeds' sources for this era, actually gives a few good examples of the variation of form in Renaissance art.
I think they've pulled inspiration for their model from artist like Rubens, who had a penchant for 'bigger' women in his paintings, but was actually painting during the Baroque era. This idea that the Renaissance featured mostly big women in its art is fairly exaggerated, and we've taken one or two examples and misrepresented the era. For the majority of Renaissance painters, accurate portrayal of human form was essential, and the women were of an average size.
Their example for Egypt also fall into this trap. Stating that Egyptian women 'are shown as slender, with high waists and slim shoulders', is just taking the period's art too literally.
My problem with the video, and its sources, is that there's no consistency. A similarly tall, slender style can be seen in Greek art from the Late Classical Period (roughly the same years as their Ancient Greek example), and yet the art has not at all been considered in their choice of model. Instead, they cite this statue as why the Greek example is 'plump, full-bodied'.
Also, not art-related, but the 'disfigured versions of men' line for the Greek woman is paraphrased from Aristotle, and was taken without context from his work, 'On the Generation of Animals'. But the writing is his personal studies of zoology and biology, and Aristotle wasn't referring to women's beauty, he was looking at it more from a scientific and philosophical point of view.
Overall, I think the creators for this video took a lot of liberties to fit their agenda, and that it lacks a lot of credibility. They do provide sources, but I think most of the information has come from cultural stereotypes and loose history.
I apologize in advance for not having excellent sources, but the whole bit on the Han dynasty is not correct. Basically everything they wrote on the Han dynasty actually describes either the Tang dynasty or modern China (Edit: I read the accompanying article and their sources on the Han are terrible. Two of them reference the same list of desired traits from the Tang, who were centuries after the Han, and one doesn't mention the Han at all). I'm not sure if anyone knows what beauty standards were during the Han.
The whole idea of "small feet" probably began in the Tang dynasty (shameless wikipedia link). There are stories claiming it originated as far back as the creation myths, but these are probably modern stories intended to justify the practice. There is absolutely no evidence that the Han engaged in foot binding.
Similarly, the "Big Eyes" thing almost certainly dates to the 20th century and no earlier. I apologize if this breaks the 20-year rule, but "Big Eyes" trend is typically seen as emulation of the West or of anime.
The "small waist" thing is very difficult to say either way. I could find no references to waist sizes, during the Tang or earlier. The few surviving statues of Han women rather effectively disguise their body type due to the preferred flowing robes, and as /u/chocolatepot mentions images of female beauty do not necessarily correlate with commonly held standards.
The Pale Skin is by far the most believable claim, but even that is dubiously sourced. Most descriptions of the fair skin traditions in the Han dynasty were written by later historians, who had an obvious bias towards applying their own standards on their past. Take this source which mentions a description of Han (ended in 220 AD) skin practices written by someone in 1792 AD. I might be missing sources, but the closest I could get to the actual Han was this short poem (~500 AD, still centuries after the Han) praising the fair skin of one woman.
So what standards of beauty did exist in the Han dynasty? No idea, I can't find any sources either way. Most descriptions of female beauty products and standards begin in the Tang dynasty.
The only internet source I can find on Han beauty is this one, which describes the "great beauty" Zhao Feiyan. Notice the complete lack of any references to her feet or eyes, and with her only defining body trait being "slender."
So I cannot say for certain that the standards for female beauty during the Han were not what was shown in the video, but there is a profound lack of sources for most of their claims. Furthermore, the beauty standards described are almost a perfect match for modern standards of female beauty in China, which makes me rather suspicious.