Room-conditions supercritical fluids?
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.