An answer to a different question on this site just informed me that it is possible to melee kill a level 3 sentry gun in TF2.
The ability to do this with any kind of consistency would make me a god among men on the casual / noob servers that I generally play on.
What technique(s) should be used to melee-kill a lev3 sentry?
Outside of an uber-charge, if you manage to reach a sentry gun before it can turn to face you, you can circle-strafe it... that is to closely run around it in a circle while facing and attacking it.
This works best as Pyro and Scout, although a Your Eternal Reward Spy with no disguise can uncloak and circle-strafe a sentry to plant a Sapper on it as well.
Note: You have to watch which way the sentry is turning and circle it in the same direction.
Note 2: It's also important to note that circle-strafing fails if the sentry is flush against a wall. However, since it's much harder for an engineer to hide behind a sentry that's against a wall, most Engineers won't build them that way.
Let's quantitatively examine the data: level 3 sentry guns top out at 216 health. Most stock melee weapons (minus the Scout and Spy) do 65 damage per hit and, while other melee weapons may also do 65, no other melee weapon does more unmodified.
So, four hits with most unmodified melee weapons will do the trick. This can take some time so I defer to Powerlord's answer in this case: circle-strafe your heart out.
Of course, there's the problem of getting to the sentry before you have the opportunity to do your melee damage. If you can't find a way to distract the sentry in another direction, your best bet is to Just Go Fast. Since we've determined that the Scout (more specifically, Atomic Punch) is a sad excuse for doing lots of upfront melee damange, our choice falls on the Demoman's Chargin' Targe.
Using the charge functionality, the Demoman moves roughly 2.63 times faster (750 HU/sec) than his base (93%) speed. That's a lot less time you're being shot at.
All those things I said about melee weapons not doing more than 65 damage don't really apply for the Demoman. Sorry, I lied. The Scotsman's skullcutter does 78 damage and it has an enhanced range (at the expense 15% speed, but that doesn't matter - we're using the charge).
So, you charge the sentry and one of three things happen:
Case 1, minicrits: Your skullcutter does ~105 damage for a limited amount of time. Not quite enough damage to finish it with two hits. Probably not an effective plan.
Case 2, just crits: Your skullcutter does ~234 damage for a limited amount of time. One-hit kill on the level 3 sentry. Probably a very effective plan.
Case 3, 50 damage + crits: You get the idea... it's even less likely that the sentry will survive after 50 + 234 damage.
In the case that we're including the Ullapool Caber (like, if this is a suicide run), not much changes. The half-charge minicrit case does ~200 damage - a lot but not enough for a one-hit kill. Crits with the Caber do 300 so that's a good play if you don't care about living through it.
My suggestion is to NOT aim your charge directly at the sentry, but just next to it. If you can time your charge right, you can finish your charge slightly next to and behind a sentry (giving you precious seconds while it turns to fire) and then turn around to smash with your end-of-charge crit - again, a single crit skullcutter hit does more damage than a sentry can handle.
Surviving the trip to a level 3 sentry isn't easy either. With a 175 base health, you can survive a little more than a second being directly shot at by a level 3 sentry, so your chances increase by being overhealed.
Oh right, no crits on buildings. I still stand by the Targe, only this time I highly recommend charging behind the sentry and slamming it with the Caber.
Realistically, one cannot consistently melee destroy level 3 sentry guns because some gun placement positions make them more or less invulnerable to melee attack.
In some sentry gun placement positions you will be unable to close for melee with a gun without the assistance of some other distraction, like another ally. This is because a level 3 sentry gun has a fair amount of pushback that will make reaching a sentry gun by normal walking, once it has begun firing, a sad and doomed process. That is unless one can get above or around the corner of sentry gun without having to walk through the sentry gun's full radius of fire.
A spy can get close, but his melee weapon is usually not the best weapon for defeating a level 3 sentry gun. The scout can drink his bonk and get close, but the pushback makes closing for melee extremely difficult. A demoman can charge in and maybe get a circle-strafe going resulting in destruction, but his weapons are not particularly suited for destroying level 3 sentry guns.
The pyro has a decent chance of killing a level 3 gun with his (or her's) homewrecker, but closing to melee range with a level 3 sentry gun is very difficult.
Ultimately the most consistent way to melee destroy a level 3 sentry gun, is to kill the engineer ahead of time so he can't heal the gun, and then attack en masse with a group of allies. The allies can distract the gun while you close for melee and finish the job.
