Python 3, 61 58 bytes

f=lambda x:0<x<=.5and f(x/(1-x))/2or x and-f(0-x%-1)-x//-1

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This answer gives a recurrence relation for the Minkowski question-mark function, so I just used the recurrence.

The input must be a fractions.Fraction object (which I believe is allowed as it is a native rational type).

Maple,  83  76 bytes

s->(a:=convert(s,confrac))[1]+2*add((-1)^m/2^add(a[j],j=2..m),m=2..nops(a));

Input and output are rationals. Calculates Minkowski's question mark function.

Charcoal, 69 bytes

ＮθＮη‹θ⁰≦↔θ⍘÷θη²≧﹪ηθ.≔Ｅ²ιζＦ²⊞υ¹Ｗ⁻×θ§υ⁰×η§ζ⁰«≔‹×θΣυ×ηΣζεＩ¬ε§≔ζεΣζ§≔υεΣυ

Try it online! Link is to verbose version of code. Takes input as the numerator and denominator of a fraction (this does not need to be fully reduced) and outputs in binary. Explanation:

ＮθＮη

Input the fraction.

‹θ⁰≦↔θ

If it is negative then print a - and remove the sign.

⍘÷θη²≧﹪ηθ.

Output the integer part of the fraction in binary, leaving the fractional part.

≔Ｅ²ιζＦ²⊞υ¹

Start a traversal of the Stern-Brocot tree with 0/1 as the lower bound and 1/1 as the upper bound.

Ｗ⁻×θ§υ⁰×η§ζ⁰«

Repeat until the input is found.

≔‹×θΣυ×ηΣζε

Calculate whether the input is below the next fraction.

Ｉ¬ε

Output 0 if it is and 1 if it is not.

§≔ζεΣζ§≔υεΣυ

Update the relevant bound.

Previous 60-byte attempt only mapped ℚ into ℤ[1/2] ∩ (1, 1):

ＮθＮη‹θ⁰≦↔θ0.Ｆ²⊞υι≔⮌υζＷ⁻×θ§ζ⁰×η§υ⁰«≔‹×θΣζ×ηΣυεＩ¬ε§≔υεΣυ§≔ζεΣζ

Try it online! Link is to verbose version of code. Takes input as the numerator and denominator of a fraction (this does not need to be fully reduced) and outputs a the binary representation of a fraction. Explanation:

ＮθＮη

Input the fraction.

‹θ⁰≦↔θ

If it is negative then print a - and remove the sign.

0.

Output the leading 0. of the binary fraction. (The output is always in the open range (-1, 1).)

Ｆ²⊞υι≔⮌υζ

Start a traversal of the Stern-Brocot tree with 0/1 as the lower bound and 1/0 as the upper bound.

Ｗ⁻×θ§ζ⁰×η§υ⁰«

Repeat until the input is found.

≔‹×θΣζ×ηΣυε

Calculate whether the input is below the next fraction.

Ｉ¬ε

Output 0 if it is and 1 if it is not.

§≔υεΣυ§≔ζεΣζ

Update the relevant bound.

JavaScript (Node.js), 75 70 bytes

f=(x,y)=>x<0?f(x+y,y)-1:x&&(x/y|0)+((x%=y)<(y-=x)?f(x,y):f(x+x-y,x))/2

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Ref

Uiua 0.16.0-dev.2, ³⁶ 24 bytes ^SBCS

+/-⇌ⁿ⊙1/2-1\+:°⊂⌊⍢˜⟜⊃◿÷±

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Takes p q on stack and outputs as a float. Both solutions compute Minkowski's \$?(p/q)\$, but in very different ways.

Explanation

⌊⍢˜⟜⊃◿÷± # continued fraction expansion
:°⊂        # take off a_0 and save it for later
-1\+       # running sum starting at -1
ⁿ⊙1/2     # 1/2^n
/-⇌       # alternating sum
+          # add a_0 back

Old solution

+÷-1⊃⧻˜⊗⍥(⊏⍏≡/÷.⊂⟜⧈+):⊞≤.⇡2⊙⊓⊂⌊⟜⊃⟜◿÷

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Takes b a on stack for rational \$\frac{a}{b}\$ and outputs a float.

Explanation

⊙⊓⊂⌊⟜⊃⟜◿÷            # stack afterwards: b  |  [b, a mod b]  |  floor(a/b)
⍥(⊏⍏≡/÷.⊂⟜⧈+):⊞≤.⇡2  # compute successive mediants, i.e. the Farey sequence of order b
˜⊗                     # at what index is a/b?
÷-1⊃⧻                  # divide by length of sequence minus 1
+                       # add back the integer part