I have thought of a situation where an emulator would help, and I am wondering what emulators exist that would fit these needs:

* Must be able to run the GOG version of Akalabeth
* Must have save state support (this is where DOSBox fails)
* Ideally, it should allow for the recording and playback of input sequences

So, any emulators that would fill these needs?

There's an unofficial DOSbox build with savestate support, although it''s not up to date- see the first entry listed under Obsolete Builds for details. Still, it might work for your purposes.
https://www.dosbox.com/wiki/SVN_Builds#List_of_SVN_Builds

You might want to look into Bizhawk for that sort of thing.

Edit: Investigating further into the TAS videos website, it appears they have their own special emulator. Seen here, on the infobox on the right.