Okay this quest is well-known. It was an excellent addition to this game back in late 2238, and still popular today. It is a very good way for newer players to realize they suck get better.
And you can solo it! I mean, in Real-Time, with the right approach.
I imagine this post will generate some replies and more help so i will begin with just a pugilist. Yes, that's right, you can solo this quest without dying even just using a Mega Power Fist, but only if you strike just the right balance between caution, confidence, skill, aggression, persistence, and patience. (I guess that's a lot). Plus, there's something more satisfying about winning fights with your fists over bursting yet another few thousand bullets downrange.
(Devs please do not mess with this quest - it's in a perfect place as a learning curve stepping stone, and is one of the ways new guys gradually get closer to TC-ready.)
The Build
So first you need the right puncher's build. ST 7, EN 10, AG 10 are the key stats, plus Heavy Handed is a must. You'll use Buffout/Nuka to get to ST 9 and Psycho to get ST 10. Max ST combined with HH is how you fight all these enemies and win. Use beer too for extra DR.
I prefer getting both Action Boys but you do it however you want. Faster punching means you can knock down multiple targets almost simultaneously. Adrenaline Rush, Toughness, Even Tougher obviously. Action Boy, Action Boy, Lifegiver, Lifegiver, and Bonus HtH Attacks. But do it however you want because it's all the knockdowns you get from ST 10 + HH that makes this work.
Prepping
Bring an extra round of drugs (or two) mentioned above. Punching is slower so it takes longer to beat this quest this way. Have some Expanded Lockpicks. If you don't have much LP skill, bring a dynamite or plastique to blast open the upper gate where the quest case is. Now look, punching is bloody work, so bring some ding-dang super stims man! You're gonna need 25 maybe.
Starting the Fight
Enter the quest location, drug up, but do not take on those outside guys yet. Run straight right and enter the cave via the hidden entrance. Take on those two enemies there. They only use power fists and you can keep them both knocked down if you're fast.
Next!
Two down! Heal up, and pick open that gate that leads to the rest of the hideout. Recharge AP (rAP) and run right up to that laser rifle mutant, land in the hex behind him - make it quick, and start punching him down. His flamer buddy will move in to help, but every time he fires you up, he lights up his laser rifle buddy too haha! Done with laser rifle, now take down that flamer mutant before he burns you anymore.
Step by Step
Four down! Feels good huh? Heal up again (use FA every time over stims when FA is ready), and rAP. Now let's get those two outside. This is trickier. They both shoot high-dmg guns. The idea here is to exit the front of the cave, and run into that little building, stand in just the right hex, and that way you only have to fight one of them at a time. This works if you choose the right hex. You can punch the first one on the way, and even the next one, but be on your way to that hex just inside the building. Using the building as a shield is the key here. (Sorry i don't have screenshot now.)
Strategy
Up to now, you've been taking them on in pairs. But next, you'll be taking on the ones in the middle, so expect at least four at a time. Whenever fighting more than 2, it's best to string out your enemies rather than let them get all bunched up. So one strategy here is to open the two doors first, that lead into the middle. That way when they start to chase you, the lead target will not get delayed as he reaches the first door, which would otherwise mean his buddies right behind him get bunched up close to him. If you leave that door closed, you'll be taking on a mass of them as if its one conglomerate enemy with too many hit points. So open those doors first, then get ready...
Guerrilla Tactics
When you upset the ones in the middle of the cave, you get four enemies chasing you. Get them chasing you and duck back toward the entrance. Take some risks before you exit - you'll learn that way, but ultimately you have to exit the cave entrance, wait just the right amount of time for them to turn around and start walking back inside, then come back, wait for the right moment and hit the straggler! If some of them keep walking away, then stick around for this fight because you can win it. If all of them turn again to chase you, then try it again - exit, wait, time it right, then re-enter.
Depending on what weapons they use, you can take on more than two at a time. The power fists rarely knock you down due to your high ST/EN, and their flamers help you more than they hurt you if you brought good armor.
Up North
Once you clear the middle area *whew! congrats!* heal up, rAP and head north. Don't take long - go straight up and hit the closest one. That will upset his buddy farther north, but it also often gets more attention! One or more of the far-right mutants will come out too, so you could run south and drag them all toward the gate for a repeat of earlier. Use the corners in the south for cover when you need to heal and rAP.
Go Try It
If you get that far solo with a MPF, then you don't need anyone to tell you the rest!
Bring a Friend
Climbing this game's learning curve can take a while. If you're not winning this solo, bring a buddy. A sniper or any single-shot helper can make a big difference. If he's a burster, be careful not to get bursted by him - the enemies will already be doing enough of that! And if you need more than a few guys to help, and even then you all have trouble...well...then you had some fun, right? And you know you need to practice because...this can be done solo with almost any tier 2 weapon. If you're not there yet, just keep climbing.
(I have died repeatedly in the past in this quest with a group of guys, even in Turn-Based, so to now be able to solo it in Real-Time should tell you it is possible. It's just a matter of honing skills and striking that balance.)