You can as long as your in the science career but once you quit it doesn't let you use it any more even if your at level 10 of the career when you quit. It'll give a message something like only a scientist has to knowledge to use it.
@Shewolf13 I need to change this statement. It seems that with the patch in March your sim can continue to make serums after quitting or retiring from the Scientist career but only if they have been a scientist sense the patch.
To test this I went back to one of my old saves before the patch where I was doing all the careers and that sim couldn't make serums even tho he had been to level 10 of the career so I had him rejoin the scientist career and used cheats to get him back to level 10 and had him quit and he could still make all the serums he had learned from when was in the the career before.
The option to do experiments was still grayed out after he quit and said only scientist could do this but he could still make all the serums and tainted serums.