The trick is simple. Write the operating system image on a pendrive instead of an SD card. Format the card just like you do for everyday use with a camera or smartphone, i.e. just create a single partition with regular FAT file system on it. Now mount the memory stick and you will notice that the first partition is also a FAT parition with a couple of different files - they are required to boot the operating system. Copy all files and folders from the stick to the card. Now edit cmdline.txt file located on the SD card (this is important, remove the pendrive before editing to make sure you modify the right file). The file contents will be similar to this:
dwc_otg.lpm_enable=0 console=ttyAMA0,115200 kgdboc=ttyAMA0,115200 console=tty1 elevator=deadline root=/dev/mmcblk0p2 rootfstype=ext4 fsck.repair=yes rootwaitIt's the arguments passed to the kernel on boot up. The root parameter tells the kernel where the main file system is located. Change it from root=/dev/mmcblk0p2 to root=/dev/sda2 and save the file on card. Connect the card and the pendrive to the Pi and turn on the power. It will load the kernel from the card, but will mount as root folder the second partition of the memory stick, containing the actual operating system (and recognized by Linux on boot up as /dev/sda2).
I used this trick to run Minibian, having a few memory sticks and just one small 128MB SD card at my disposal. If the Pi doesn't boot from /dev/sda2 make sure you don't have any other usb storages connected, otherwise try a different device instead of /dev/sda2, like /dev/sdb2, /dev/sdc2, etc.