Yes, in theory it is possible. The Internet Archive is in the process of doing so, and has archived (not backed up - there is a difference) 452 billion webpages, as well as movies, music, books and other media. However, this is only ~3.16% of the 14.3 trillion web pages in existence as of June 2013.
In 2013, the Internet was estimated to be about 672 Exabytes or 672,000,000 Terabytes large, assuming 1EB=1000PB, 1PB=1000TB and 1TB=1000GB. To put that into prespective, you'd need 336,000,000 2TB HDDs to hold all the data, assuming no HDDs fail, 100% of the space can be used and the internet doesn't grow larger. Where you'd store them is another matter. Assuming the size of each HDD is 3.5" by 6", you'd need a building with at least 4,083,333.33 cubic feet of space to store the HDDs, without even considering the space needed for additional equipment, power supplies, ventilation or even cabling. With all that, it would be closer to 4.5 million cubic feet.
Assuming you'd use Seagate 2TB Barracuda HDDs, priced at $58.65 each, you'd be looking at spending $19,706,400,000 (19.7064 Billion Dollars) on HDDs alone. Then, assuming you have an internet connection with a bandwidth of 100Gb/s, and that you can use the entire bandwidth for the download, it would take approximately 1830.4274 years to complete.
And all this is based on data that is two years old; the internet is certainly larger now.
So yes, you can download the internet, but you'd need a vast infrastructure, extremely high budget and superb internet connection to do so.