Cement is 150lb/ft3 . You'd have to build the cement building as a bunker, dropping a lid sarcophagus style, so that the trailer doesn't bear any weight.
You don't need a 300' hole, this isn't a bomb shelter. As a fallout shelter, 30' of earth is solid, in most scenarios 10' is sufficient.
Air filtration and ventilation will be important. Most radioactive particles are relatively large, but still elemental so HEPA isn't good enough, try to get a CO2 scrubber system if possible. Otherwise, you'll be relying on the ventilation pipe shape to keep most (at least the heavier) particles out.
Have iodine tabs handy (or just get in the habit of using iodized salt on everything).
Have food.
Don't forget the can opener.
Have fresh water. This is too easily overlooked. (Also, another reason not to go too deep.. wherever your local water table is, you could quickly find too much water is a problem).
Compost toilet, or you'll wish you did.
Have a plan for power. This whole setup needs to be 100% off the grid (if connected during a blast, everything will get fucked). You can't use a gas generator in you living space, and you're not going topside for at least 4 days so clearing the radioactive dust off solar panels isn't happening (though you might get lucky and have the black rain). A VAWT could work. HFC is nice if you have the money but I get the sense this is an "on the cheap" thing. You could even have a bicycle generator (gives you something to do and keep your legs fit while down there). Make sure your lighting is energy efficient; incandescents don't last long on battery.
Communication.. Expect modern digital infrastructure to be kaput. There might or might not be a radio station running.. have an antenna run to the surface matched to whatever you're using. Your best source of info will probably be on shortwave. (side note, do not leave antenna hooked up to radio before use.. the antenna will pick up EMF from a nuclear blast and anything attached could get fried.)