I agree with the sentiments above. Should you, no. Could you, yes.
I have 6 tangs in a 300, 8' tank. I really have no problems other than the purple tang going after my two yellow tangs once in a while. Others include: large powder blue tang, large blonde naso tang, and a a medium regal blue tang.
So why do I say no: mostly the blonde naso tang. I love mine, maybe my favorite fish. It's beautiful, has a great disposition, always out, but he is huge. In 2 years it has gone from a 2.5" fish to a 9" fish plus streamers (that is not an exaggeration). I'm worried I may have to re-home him someday, which is not fair to the fish. I've seen huge sailfins and I would say the same about them, and this is in an 8' tank.
If you want to keep 4 smaller tangs, that would probably work. Start small, feed lots of nori (I give 2.5 sheets per day) + mysis in selcon, lots of filtration/ nutrient export, have rockwork for hiding but a stretch of open lane for fish swimming, and have small peaceful fish to distract.