There is absolutely no problem for the society to survive in your scenario. Killing half of plants and animals will barely be noticed. We kill more than that all the time. Most species will recover within a generation or two. It may take a while for long living species like Oaktree or Blue Whale, but that's not a serious concern.
As for human, yes, there will be no problem either. Relatively speaking. According to different estimates, between 20 and 80% of jobs are unnecessary anyway. Many companies will struggle in the new situation and may collapse when their services are no longer a priority (e.g. producer of reality TV?), but their more productive employees will find jobs in essential businesses.
Fewer people means lower demand for services as well. Population has doubled in last 50 years. As you may imagine there was a society 50 years ago. 998
As in the other answer, we had large scale depopulating events before. It's not unheard of for a country to lose 20% population in a war, even in recent times, for example Poland in World war 2. They not only lost a fifth of their population, including most of the educated elites, but were also physically devastated by war. Warsaw was nearly obliterated with around 90% of buildings destroyed. After the war they were trapped behind the iron curtain and cut off from international trade or cultural exchange and yet managed to survive as civilized societies and rebuild from ruins. In your scenario most of infrastructure is intact, at least initially, so your society can absorb much higher loses.
The sudden drop will be drastic but will not destroy our civilization. The old, inefficient power plants will just be closed as the demand for electricity halves. The good nuclear plants that lose half of their staff will halve the holidays for the survivors for few years and recruit some staff from closed plants and recent graduates to fill the gaps. At the moment half of STEM graduates work in unrelated jobs. Instead of joining investment banks they will get productive and fulfilling jobs in their field. No problem at all.
TL,DR Apart from initial panic, there won't be a long term danger to the society if half of the population disappear. On the contrary, if you're one of the survivors and don't die in some riots in the immediate result of the event you may even be better off than before. 998