You can sign up for one of the free cloud storage accounts. The problem is that most of them that are free are not that large. Once you start searching you are going to see what I mean. You can certainly open multiple free accounts. Just keep in mind that the time necessary to copy all of those files from your computer to the cloud storage is still going to be a fairly slow process that will be dictated by how many files there are, size of those files, the speed of your computer, speed of your internet service provider upload speed, and speed of the website you are copying to.
Me, personally, I would buy an external hard drive and copy the files to that. You can buy an external for less than $100. You will still have the time issue of copying from one to the other, but with the magnitude of files you are talking about, you are going to have that issue no matter what type of storage you use. Plus, many of the files to which you refer can remain on the external as a permanent storage. For ex there is no need to have to copy the videos or photos, or even the music, back to the new computer hard drive as each (which is your size monster anyway) can remain stored there. So in this case you'd only have the time issue of ONCE for copying from the computer to the external and not have the issue of copying back to the hard drive.