Is it ever ac­cept­able to write slashes be­tween mul­ti­ple but sep­a­rate pre­fixes?

The key point to always remember is to make your communication unambiguous and easy to understand. This takes priority.

The slash is ill-defined in English, and thus frequently achieves confusion.

I've seen various writings where people have used the slash to mean any of "or", "and", "both", "either", "a combination of", a hyphen, etc.

I've had occasions where someone has used a slash, and I've genuinely had no idea what they meant. Despite rereading the paragraph several times, I've had to ask the writer to explain what they meant.

Using the slash slows down the reader, as they try to understand what you actually meant. Does "bio/tech­no­log­i­cal" mean "bio and technological", "bio or technological", "a combination of bio and technological", "biological and technological", "biological or technological", or something else? (When I first saw what you wrote, I thought that you meant "bio-technological", i.e. technology related to biology.)

For this reason, I would avoid using the slash except in the rare cases where its use is both widespread and obvious. (This is rare enough that I'm struggling to think of an example.)

One alternative is to use the hyphen. An example might be "uppercase and lowercase", which can be written as "upper- and lowercase". In your case, this would result in "bi­o­- and tech­no­log­i­cal". But this is somewhat forced, and thus causes the reader to have to slow down or pause, which reduces its readability. Therefore, I advise against this (although it's preferable to using a slash).

The best solution uses the conjunction that you mean. In your case, "bi­o­log­ical and tech­no­log­i­cal" does the job: it's unambiguous and easy to read, and it doesn't slow down the reader. It allows the reader to concentrate on the message, not on the language.