The original solution to this was BinHex. It hex-encodes a file so it is ASCII and can be transmitted over practically any protocol. I think you could use an old version of Stuffit in your emulator to encode your .sit file as binhex. You get a file with the .hqx extension as a result that can be decoded in stuffit. There was originally a program called BinHex that just did the .hqx encoding.

The first Mac users encountered this problem when downloading programs from CompuServe. They had to download BinHex from CompuServe, but they couldn't decode it without having BinHex. The solution, as I remember, was a MacPascal program which was downloaded as text, but when executed, created a working binary version of BinHex.