Chemistry - Room-conditions supercritical fluids?
Solution 1:
As you can see on this data page on Wikipedia helium-4 has critical pressure of 2.24 atm, but helium-3 has 1.13 atm, only slightly above atmospheric pressure. Hyperbaric chambers work with 1,3-1,5 atm, so you could try there.
There is, unfortunately, question if it has sense, as you wouldn't notice any serious difference in properties between subcritical helium-3 and slightly above critical point in pressure, but so far above in temperature. It would still be lighter than air, for example.
I don't think that much higher pressure would be a huge problem - the highest pressure survived by human is supposedly 54 atm - so says Wikipedia, but I didn't find proof. If there is a compound with critical parameters sufficiently near standard, you could touch liquid-like, not gas-like supercritical fluid.
Ionic liquids have relatively low critical pressures (even only 14 bar), but too high crit. temps (source). On the other hand ethane's crit. temp. is below 30°C, but pressure over 48 atm. Maybe there's sweet spot somewhere, but it would be still high pressure and may be quite hot or cold.
Solution 2:
A number of long carbon chain molecules with large numbers of fluoride atom, have $P_c < \pu{10 atm}$ and $T_c < \pu{1000 K}$. That is within the reach of a good bicycle pump and a burner. In particular $\ce{C12F26}$, $\ce{C15H4F28O}$ and $\ce{C20F42}$:
\begin{array}{lll} &\ce{C12F26} &P_c = \pu{912 kPa} &T_c = \pu{417 K} \\ &\ce{C15H4F28O} &P_c = \pu{784 kPa} &T_c = \pu{701 K} \\ &\ce{C20F42} &P_c = \pu{463 kPa} &T_c = \pu{700 K} \end{array}
The critical point of a fluid is determined by its van der Waals constants. For a fluid to be critical at STP it needs these constants to have the values $a = 25$ and $b = 0.003$ in SI units. The value of $b$ corresponds to a molecule with around 300 atoms, but the value of $a$ is outside the range of published compounds.
However, something compounds like $\ce{C100F202}$ could be critical at around ordinary pressure and an easily achievable temperature, i.e. it might be possible to get a critical fluid by heating some goop in an open saucepan. It also seems that fluorocarbons tend to be stable and inert, so they may be safe enough to handle.
@Diracology and @Floris provide an in depth exposition of the relationship between critical points and van der Waals constants in their answers to my Physics.SE question Fluids with critical point at ordinary temperature and pressure. Their answers with internet search of van der Waals constant data lead me to start looking at at large fluoroacarbons. The ones listed above are the largest I could find data for. It should be possible design more suitable molecules computationally. See Computing van der Waals constants from molecular structure.
Solution 3:
If exceeding the "liquid–liquid critical point" for a solution suffices, then perhaps you could survive contact with methane hydrate for a short while. Though He is supercritical at ~two atmospheres, running you hand through it would be a chilly experience at about 5 K.
Solution 4:
Quick googling got me to $\ce{CF3Cl}$, that has cr.p. at $\pu{28.8^\circ C}$ / $\pu{38.6 bar}$. $\ce{C1}$-$\ce{C2}$ freons look to have critical point in $\pu{30 .. 60 bar}$ / $\pu{0 .. 100^\circ C}$ range, so some of them probably are survivable. However, some compounds of this class are used for general anesthesia, so $\ce{He/O2}$ mixture respirator would be required.