The parts of a Tesla coil most likely to kill you run at DC or mains frequency, sit inside the cabinet, and give no warning. They are the supply, the primary circuit and the capacitors. The site's safety guide ranks them the same way, well ahead of the discharge everyone is watching.
Skin effect is the usual reason given for surviving a streamer hit, and it is wrong. Skin depth goes as one over the square root of conductivity times frequency. In copper at 400 kHz it is 0.103 mm (the primary is an impedance). Tissue conducts something like a hundred million times worse, so at 400 kHz its skin depth is about 0.8 m to 2.5 m for conductivities of 1 down to 0.1 siemens per metre [derived by scaling the copper figure]. The conductivity is our assumption, but at any value in that range the current goes through the middle of you.
What changes with frequency is how much the current excites nerve and muscle. At mains frequency it drives them directly, giving the grip you cannot let go of and putting a heart into fibrillation. At hundreds of kilohertz the same current mostly heats, so a hit burns rather than fibrillates (the pupman safety document, with its limits discussed below). Nothing keeps the current near the surface, so the burn can be deep and worse than it first feels. None of this makes contact with a running coil safe. If the coil strikes something on the mains side while somebody is in contact with a streamer, the ionised path connects that person to the mains, and the mains kills them.
The bus bank
The shock physics above rests on one source, the Tesla Coil Safety document on pupman, contributed by Chip Atkinson, Ed Phillips, Mark S. Rzeszotarski and R.W. Stephens. It gives the mechanism in one line: "Higher frequencies are less able to cause these involuntary contractions." It also warns that the heating from a large coil is mostly internal, which a surface-current story would not predict. A few lines earlier it repeats the skin effect claim anyway, with a figure of about 500 kHz attached and an admission that it has little data between 50 and 250 kHz, the band coils actually run in. The mains path through a streamer is its sharpest warning, and a grounded strike rail lowers the odds of it without ruling it out.
That source leaves two gaps. First, we have no source for where the roll-off happens. The one line is from an amateur document that disclaims its own completeness, and it puts no frequency on it. A threshold is quoted in the wider literature, but we have not sourced one, so no figure is given here. Second, the line is about involuntary contraction, the grip. Extending it to fibrillation is our own step, on the grounds that the heart is muscle and the mechanism is the same, and no source here says it of fibrillation. Take the mechanism as the reason the arc is not first on the list, and never as a licence to touch any part of the machine.
The bus bank itself is thousands of microfarads at a few hundred volts. On 230 V mains that is about 325 V behind a plain bridge and about 650 behind a Delon doubler. The bank on the machine this site works from is 9900 µF, and every figure in this section comes from the bus bank, which sizes the part properly. That is a lethal store of energy, and it is still there after you switch off.
The bleeder has to be sized. Across that 9900 µF, 6 kΩ takes two and a half minutes to fall to 50 V and dissipates 70 W. 2.4 kΩ takes a minute and dissipates 176 W. Both are serious resistors and the wait is long either way, so the machine also needs an indicator.
A bright LED with a dropper resistor shows that the bank is up and bleeds it slowly at the same time. A neon makes a useful backup, but mind which neon voltage you are thinking of. It strikes at about 90 V and does not go out until nearer 60, so a dark neon means under sixty, not under ninety. Sixty volts across thousands of microfarads will still hurt you badly, so a neon cannot be the only indicator.
Precharge protects the machine rather than you. A cold bank at switch-on is a short circuit across the mains, so a resistor goes in the AC line before the rectifier and a relay shorts it out once the bank is up. Work the delay out from the time constant. Sixty ohms into a single 9900 µF bank behind a plain rectifier is 0.59 s. The same sixty ohms behind a doubler is 2.38 s, four times as long. The doubler's capacitors are in series and each side is fed by its own diode for half the cycle, and the two effects multiply. Three seconds of relay delay is only 1.26 time constants in the doubler case. The bank is then 72 per cent charged when the resistor is shorted out, and the rest of the charge arrives with nothing limiting it. Set the delay for the machine in front of you.
The frame under the primary
A metal frame under the primary is a one-turn secondary of that primary.
Break the loop with nylon bushes, washers and Kapton, so that each upper rail is grounded at one end only, and take the strike rail to RF ground on its own wire rather than through the frame. The numbers are in grounding a Tesla coil, including how wide that break has to be. It is more than the two millimetres it looks like.

The secondary's ground
Use a counterpoise instead, with a short lead to it. That has its own piece.
The current transformer nobody thinks about
Which of the two burdens takes an ordinary probe, and which needs a differential probe, is in what a DRSSTC current transformer is, and what it senses.
The tank, while the coil is off
The tank capacitor has been at kilovolts in a resonant circuit, and it holds that charge like any other capacitor. Treat it as charged and short it on purpose, with something insulated.
A bleeder across the tank costs nothing in performance, so there is no reason to leave it off. The site's safety guide asks for roughly 1 to 10 MΩ across every high-voltage capacitor. At its working frequency the tank capacitor's reactance is the primary circuit's characteristic impedance, 37.3 Ω on the coil this site works from, from a 16.96 µH primary and a 12.2 nF tank (the primary is an impedance). One megohm in parallel with 37.3 Ω of reactance adds 1.4 mΩ of series resistance [derived, X²/R]. The tank loop itself is tens of milliohms, about 90 mΩ on the coil fitted in SPICE in dead time as a fraction of the period.
The same resistors drain the bank fast. That 12.2 nF is 99 parts of 10 nF, nine in series by eleven in parallel (the MMC is sized by current). With 1 MΩ across each part the time constant is 10 ms, for one string and for the entire bank alike [derived, since the resistance and the capacitance divide the same way]. Use the shorting stick anyway. A cracked resistor or a lifted lead looks exactly like a working bleeder.

And the thing that catches people who know all this
Most damaged equipment, and a fair number of injured people, come from the measurement setup rather than the coil. A scope's ground is bonded to mains earth and a coil's power section is not. Scoping one without killing anything covers it. Read it before the first power-up.
These are the minimum. The site's safety guide goes further, and none of it replaces the habit of treating every capacitor in the cabinet as charged until you have shorted it yourself.