Pigeons are a lousy answer. It takes way too long to train them to memorize the file, and they usually make significant mistakes when asked to recite it back. :-)
Seriously, you never said machine B was more than ten feet away, and if all it takes it ten transport mechanisms to count as ten distinct answers, then the question can be answered trivially. ("Send a floppy by car/bus/train/pigeon/horse/elephant/ etc")
Here's my list:
Retype it, using a hex editor if necessary.
Put it on floppy/CD/zipdrive.
Ethernet.
Print the file, and scan it back using OCR. Use something like a uuencoding if necessary.
Nullmodem cable across the serial/parallel port.
Modem.
Text to speech => tape recorder => voice recognition (like a modem, but more inefficient).
Power down machine A, yank the hard drive with the file on it, re-install in drive B.
Print bar codes & read back with a bar code reader.
Hot-swap raid drives from A to B until the file is copied over. Keep replacing drives on A so that the system never goes down.