If I stood next to a piece of metal heated to a million degrees, but in a perfect vacuum, would I feel hot? A friend of mine told me that if you were to stand beside plate of metal that is millions of degrees hot, inside a 100% vacuum, you would not feel its heat. Is this true? I understand the reasoning that there is no air, thus no convection, and unless you're touching it, there's no conduction either. I'm more so asking about thermal radiation emitted by it.
 A: The other answers provide good explanation why your friend is wrong in this case. I just want to point how you could both easily reach the same conclusion without knowing much of the physics involved:
The surface of the Sun is about 6000 degrees (Celsius and Kelvin). It is separated from you by 150 million kilometres of vacuum, yet you can clearly feel it. It follows that you could feel higher temperatures too, up until the point where you couldn't feel anything at all.
A: Thermal radiation would indeed be an issue, but there are a couple of interesting facets of this question and its answer that are obscured by the hyperbole. It's instructive to peel away the hyperbole to learn more.
First of all, "millions of degrees" is not compatible with "metal" in any familiar sense. Iron boils at 2862°C. Tungsten melts at 3422°C and boils at 5930°C[1]. At millions of degrees you would have an expanding plasma ball competing with its own thermal radiation to explode and kill you. We could postulate that something confines the plasma, and in that case the thermal radiation would cook you in short order, as explored in other answers.
However, I think your friend may have been thinking of a very real phenomenon that often gets obscured by introductory physics curricula.  I don't see it mentioned here, but it has literally and metaphorically burned many people, so it's worth re-casting the question in order to highlight this phenomenon.
"If you wave your hand near a block of 660°C Aluminum, just below its melting temperature, do you feel the heat, assuming convective heat transfer is negligible?"
We are familiar with hot objects in everyday life, and we intuitively expect hot objects to radiate heat. The Stefan-Boltzmann law tells us how much power per area a blackbody radiates, and many objects in our everyday lives are decently approximated by blackbodies. Under the assumption that the Aluminum behaves like a blackbody -- which you should now be highly suspicious of -- you might intuitively expect to feel approximately the following power/area of radiated heat when you wave your hand past:
$$
\frac{P}{A}=\sigma T^4\approx (5.67 \cdot 10^{-8})(273+660)^4 \approx 4.3 W/cm^2
$$
You would feel only 3% of that. You might erroneously assume that the Aluminum has a low temperature, touch it, and burn yourself. Many have.
The reason is simply that many materials under many conditions don't behave like black bodies. Aluminum is a notorious outlier. The ratio of actual emitted thermal radiation to black body radiation is called thermal emissivity and varies quite a bit for different materials, surface finishes, and so on:
https://en.wikipedia.org/wiki/Emissivity
In the lab, this has practical consequences. You can't read the temperature of shiny metallic surfaces through a thermal camera because those surfaces will behave like mirrors, not temperature-indicating glowsticks. You can fix this problem by adding little black patches to any shiny parts you need to measure.
I startle myself at least once a year by assembling a circuit, watching it through a thermal camera for the first powerup, reaching over to turn on a power supply, and jumping back at the sudden jump in temperature due to seeing my arm's thermal reflection in the shiny components. 
[1] Taken directly from the wikipedia pages for Iron and Tungsten. I believe these temperatures assume vacuum but didn't verify that. Regardless, I wouldn't expect P=1atm to fundamentally alter the discussion.
A: You would feel its blackbody radiation as it is an EM wave and does not need a physical support to propagate itself. Also, "100% vaccuum" is not rigorous definition of the state of your system.
A: Your friend is completely wrong. Consider the following things:


*

*The temperature that you are talking about is very high, no metal would be in a solid state at the temperature you are talking about. So, before your plate reaches millions of degrees, it would have melted long before.

*Your understanding is correct in terms of thermal radiation. The radiation of Sun reaches Earth and there is a vacuum between. So, if you have an object as hot as you are talking about it will emit thermal radiation energy per unit time as per the Stefan-Boltzmann Equation. And remember, the rate of emitted radiation is proportional to the fourth power of temperature, so doubling the temperature would increase the rate by 16 times. You can calculate the energy reaching per unit area of your skin and find out what will happen!
A: 
I'm more so asking about thermal radiation emitted by it.

Here's a quantitative estimate.
Suppose that the hot plate remained intact long enough to do the experiment. For a rough estimate, we can treat the hot metal plate as a blackbody. According to Wien's displacement law, the electromagnetic radiation emitted by a blackbody at temperature $T$ is strongest at the wavelength 
$$
 \lambda = \frac{b}{T}
\quad
 b\approx 2.9\times 10^{-3}\ \mathrm{m\cdot K}. 
\tag{1}
$$
The total power emitted per unit area is given by the Stefan-Boltzmann law
$$
 \frac{P}{A}= \sigma T^4
\quad
 \sigma\approx 5.7\times 10^{-8}\ \mathrm{\frac{W}{m^2\cdot K^4}}.
\tag{2}
$$
For $T=10^6\ \mathrm K$, these estimates give
$$
 \lambda\approx 2.9\times 10^{-9}\ \mathrm m
$$
and
$$
 \frac{P}{A}\approx 5.7\times 10^{16}\ \mathrm{\frac{W}{m^2}}.
$$
This wavelength is in the X-ray range, and this power level is more than a trillion times the power a person on earth would receive from the sun if there were no clouds and no air.
Would you feel it? I'm not sure. Probably only very briefly.
A: 
A friend of mine told me that if you were to stand beside plate of
metal that is millions of degrees hot, inside a 100% vacuum, you would
not feel its heat. Is this true?

Not true.   Heat can be conducted through physical medium, or it can be transmitted as electromagnetic radiation, specifically the infrared waves, which travelled through the vacuum.
A: Actually, your friend is probably right but for the wrong reason.  That much energy is going to fry you in very short order--and will probably kill the nerves before they can say "hot!"
Remember, energy goes at the 4th power of temperature.  100x the temperature of the sun equals 100 million times the energy.  There is no question that's enough to kill you very quickly, the only uncertainty I have here is whether you will perceive anything before that happens.
A: The black body answers are fine, but I would like to point out that no one has accounted for the amount of material present. If you had a metal gas with 100 atoms obeying a Maxwell-Boltzman distribution at the stated temperature, you would feel nothing. 
