Actually, it's not that inconceivable!

AI-powered computers are already able to pick apart an audio recording into individual voices and instruments, allowing engineers to hear the most important parts in isolation. The actual software is of course a well-guarded business property, and not something you get for free, but do yourself the favor of visiting e.g.
Splitter to hear some examples. Not perfect yet (you'll certainly hear noise, artifacts and "bleed" between tracks), but I'm quite sure that subsequently converting each part to MIDI is within reach, technologically speaking.
There are a few websites that use less sophisticated means to perform online conversions from audio to MIDI for free. The technology behind these online converters are literally decades old, though, and definitely works best on very simple recordings containing as few instruments as possible (e.g. Classical piano or guitar pieces). As soon as the complexity increases, the output becomes less predictable. The resulting MIDI files often contain a lot of stuff you wouldn't expect to hear (or see when opened in a sequencer). The explanation lies in the fact that older algorithms focus purely on pitch and spectral information. There are several potential problems tied to this fact:
- Natural harmonics from the main instrument will often be misinterpreted as additional notes, high above the actual notes.
- Prominent effects, such as delay or distortion, will also be interpreted as additional notes.
- Percussive elements and noises are notoriously difficult, and a single drum hit is often rendered as a tightly-spaced series of notes.
- Sliding notes and vocals - in fact any sound that can't be defined as a single note - will be treated as a series of notes.
The list goes on, but we can already see that it fits very poorly with most modern genres of music, where these things occur all the time. As an example, i uploaded an MP3 file of The Hooters' "Karla with a K" to
BearConverter. In the MIDI file I downloaded afterwards, the mandolin intro is handled relatively well. When the rest of the band joins in, however, things quickly fall apart, and the song ends up sounding completely chaotic in places.
Last but not least, the websites are not able to detect what instruments are actually being played, so the output file is usually just a bunch of piano tracks, that try their best to replicate whatever is heard in the original file. Therefore, don't expect a polished MIDI file, ready to load into your favourite arranger keyboard. Depending on the original recording, you'll probably have to spend quite some time working on the MIDI file in a sequencer, in order to arrive at something worth listening to. At least for now.

- H -