Easiest way to remove files

Time Machine does back up the files in an a directly accessible format (for Macs that run Leopard or later). However, its usefulness is in being able to restore files from different saved backup states (go "back in time"). For example, I can restore a version of a file from six months ago that is different from the current version of that file. In your case, you just want to create an archive of the last (current) version of all the files, so using Time Machine is not necessary. You can manually copy the user data files from each Mac's storage drive(s) to the archive drive. OR you can create a disk image of each "volume" (disk) and save the disk image file on the archive drive. The disk image can be mounted and accessed, like the original volume. Manually copying of only the user data saves storage space (because you are excluding system and app files). A disk image preserves everything, as it was on each storage volume.

What are the "5 desktop computers" specifically? If any are much older Macs, you may have "interface issues" to overcome. For example, you cannot connect a USB external drive to a Mac that does not have USB ports.

How do you intend to access these files in the future? That would affect the best way to store them.

For example if it is Windows, then you would want a Windows compatible file system format, such as "exFAT". If another Mac, then the Mac native file system format would be best.

Once you have that decided, you can create a directory structure on the external disk, such as Mac1, Mac2, Mac3, .... Then drag and drop the folders of interest to the respective folders associated with the Mac the data is coming from.

Also, I would strongly suggest 2 things.

A) use multiple external disks with redundant copies of the data, as you do not want to trust the data to just one device.

B) I would try to keep one of the disks in service, as magnetic data tends to drift if left idle. In fact I would make a new copy of the disk every year or so, just to make sure the magnetic fields are refreshed (disk cloning software can do this easily enough; for Mac I would use Carbon Copy Cloner or SuperDuper; for Windows, I do not know what is the best offering, but it does exist).

You might even consider also paying for an on-line backup service, such as Dropbox so that you also have a professional service keeping a copy refreshed (BUT DO NOT put 100% trust in an on-line service, as business make mistakes or go out of business, so you want your own copies too).