Irrational packing of Euclidean spaces (with no gaps).

This appears to be a new question on MSE. The only post on here after a search using the string irrational "packing" does not mention (explicitly) what I have in mind. A Google search doesn't return anything specific worth noting to an amateur like me. Finally, MO is empty in this regard too (but that's without looking into the minutiae of the posts that show up).

The Question:

Is it possible to pack $$\Bbb R^n$$ for some/each $$n$$, with copies of a regular shape and no gaps, where each copy is of a distinct, positive irrational defining length?

Thoughts:

I'm guessing the answer is "no". Why? I don't know: pure speculation. It just seems typical of the behaviour of irrational numbers.

Having said that, though, it's just occurred to me that $$\Bbb R$$ can be packed with line segments, each of a distinct, (positive) irrational length. This seems obvious. List the line segments (assuming countably infinitely many is sufficient); spiral out from zero and leave no gaps.

I'm reminded of squared squares for when $$n=2$$. They give me confidence that the $$n=2$$ case is possible. Multiply a squared square by an irrational number. Build a bigger squared square from it by keeping the ratios between the square side lengths the same as the original squared square. Carry on ad infinitum.