Computers can be booted over a network, through a USB drive, or even off of a CD or DVD. When you attempt to run a computer without a hard drive, you will often be asked for a boot device. From there, you can select to boot from an optical drive or a USB storage drive. If you're planning on running a live system, things will be pretty easy. For example, you can use Rufus to make a portable version of Windows for your on-the-go needs, thus circumventing any necessity for hard drive installation. There are also tons of versions of Linux, with the most popular one being Ubuntu, which comes by default as a live version of the OS so you can use it without installing it at all. In short, yes, it is entirely possible to use a computer without a hard drive.