Is there any bijection from $[0,1]^3$ to $[0,1]$? How can I construct it?
If there exists a surjection between $A$ to $B$ and a surjection between $B$ to $A$, then there exists a bijection between $A$ to $B$. In your case, space filling curves are surjections from $[0,1]$ to $[0,1]^3$. It should be easy to find a surjection going the other way.
I'm not sure if this is correct, but I like the idea and would like to see how it is broken if it is.
Given any real number you can express it uniquely using its canonical continued fraction. So from three numbers you can produce three sequences. You can interleave these three sequences ($s_1, t_1, u_1, s_2, t_2, u_2, \ldots)$ and evaluate it to a real number.
Perk: This is constructive for three rational numbers.