spacetime - If Space is continuous, why can't we go below Planck Length?

I am not talking about any other attributes of particles, vacuum etc ruling out Uncertainty Principle thing. If talking about pure Space (which is continuous, not discrete, cf. e.g. this Phys.SE post), why is Planck Length lower measurement limit?

Update:
I want to avoid Uncertainty Principle at all cost. After John's answer, I am clarifying my question with abstract Space notion. I have pure Space, but no Vacuum to rule out any Quantum Fluctuation to create energy based on Uncertainty Principle.

New related question: Is lower measurement limit due to Quantum Fluctuation of Vacuum? Or, there's more to it?

Answer

There are several duplicates of your question, but they have all been marked as duplicates of How to get Planck length, and I don't think this really answers your question. Actually the best answer is by Ben Crowell to one of the duplicates, Why is the Planck length the shortest measurable length?, but Ben has deleted his answer presumably when the question was flagged as a duplicate. I'm unsure about the ethics of quoting from a deleted answer, so I'll sketch a brief explanation here.

The uncertainty principle states:

$$ \Delta x \Delta p \ge \frac{\hbar}{2} $$

So if we try to focus any measuring equipment down to some small distance $\Delta x$ we pay the price of increasing the momentum uncertainty of our system. But momentum and energy are linked by:

$$ E^2 = p^2c^2 + m^2c^4 $$

So to focus down increases the energy uncertainty of our system. As we go to smaller and smaller $x$ the energy uncertainty gets bigger and bigger, and this means the energy density rises. When we get down to the Planck length we find the energy density uncertainty has got so big that it creates a black hole with an event horizon radius of (around) the Planck length. Now we have a problem because we can't measure anything inside the event horizon of a black hole.

This means it's impossible to measure any distance less than (around) the Planck length. If we try we actually just make the black hole bigger and our resolution gets worse not better.