It would help if you told us which MacBook and what year it was made.
The Mac can boot and run from an external (USB) drive as easily as from the internal drive.
BUT... you need an external drive that IS BOOTABLE to the finder.
IMPORTANT QUESTIONS:
Can you boot to internet recovery?
If not, do you have access to ANOTHER Mac that works?
IF you can boot to internet recovery, you could do this:
(You'll need either an external drive, or perhaps an external USB flashdrive of 16gb or greater)
- Boot to internet recovery
- Connect USB drive
- IF the drive isn't formatted to Mac OS extended with journaling enabled, do this:
- Quit the OS installer, open Disk Utility
- ERASE the USB drive to Mac OS extended with journaling enabled. Give it any name you like
- Quit Disk Utility and re-open the OS installer
- Be careful to "aim" the installer AT the external drive
- Try to install that way.
When done, the new install on the external drive should boot, and you can set it up.
Note: when you boot a Mac with an external drive attached, the best way to do it is to:
- press the power on button
- IMMEDIATELY hold down the option key and KEEP HOLDING IT DOWN until the startup manager appears
- Use the pointer to select the external drive, then hit the return key.
(You can also set an external drive to ALWAYS be the boot drive in the startup disk preference pane)
Once you get "bootable to the finder" that way, you can turn your attention to the internal drive.