Is it possible to build a DIY RLC circuit for a classroom demonstration?

I would see what variable resistors were available and using those ranges do a spreadsheet to calculate results of cycling through the resistor range with known values for L and C.

Then you can decide input voltage etc - will they be able to touch it? Safety?? 12V or 120V or 1200V?

Sure! An LC tank connected to an antenna and coupled to a diode detector and an AC microvoltmeter - ideally one with a needle - will be fantastic. For the UHF frequencies (around 100MHz) you can build it with household materials with relative ease, although you will want to characterize your build process ie. use an LC meter to see how close are your homemade capacitors and inductors to theoretical values derived from their geometry and material properties.

Ordinary aluminum foil and polyethylene food covering wrap is good enough to make fairly decent capacitors in the ranges needed (1-100nF). Any sort of wire will work for the inductors although you may need ferrite slugs/cores to make it less painful to put together. In all cases you should start from first principles, look up material properties, and figure out dimensions/numbers of turns before you start making anything.