You can, yes.
This process assumes you are installing from a live cd. While a live usb should work fine as well, the cd option is theoretically the safest, as there is no chance of overwriting the cd during the partitioning.
I recommend you start off by disabling your internal HDD in your BIOS first, as this makes sure there is no chance of accidentally overwriting your internal partitions. Also, the partitioning step of the Ubuntu setup will be much easier, since it will only detect the USB drive. With other words, it's best to make the USB drive the only storage device present on the machine during the installation.
Next, boot up the live cd and initiate the installation as usual. Make sure you choose "use whole disk" if you disabled all other storage devices, otherwise you will have to do manual partitioning.
In the last case, create an ext4 partition on the USB stick (make a partition table if there isn't one) and, if necessary, a SWAP partition if you intend to run heavy applications. Set the mount point to /. don't touch the other storage devices and their partitions!
When the setup asks for the bootloader location, choose the device name of your USB drive. This can be
/dev/sdb, so on, but don't choose a partition (e.g.
Wait for the installation to complete, then reboot. Make sure your machine boots from the USB drive. This can be made sure either from the boot menu (usually esc or a function key) or from the bios, where the boot sequence can be altered.
If everything went okay, Ubuntu should boot from the USB drive. If GRUB shows up, choose the first option.
Lastly, run the following command from your freshly installed Ubuntu desktop:
sudo chmod -x /etc/grub.d/30_os-prober
This makes sure that update-grub does not detect any other OSes that may be present on the system, as they do not matter for your Ubuntu USB drive installation. Also, when you boot up your USB drive from a strange computer, the OSes on its internal drive will be included into GRUB when a kernel/grub update occurs. This is unwanted.
Also, make sure to turn back on your internal storage devices from your BIOS.