I understand that reverse-biased zener diodes don't conduct at all until they've reached a certain breakdown voltage, after which their voltage drop remains constant. I want to use this property in a circuit, but I don't have any zener diodes. Can I emulate one with just resistors, transistors and capacitors?
The 'amplified diode' circuit below is used often to bias the output stage of class B amplifiers. It can be tuned by resistor selection to any voltage, and you can use a potentiometer to make it adjustable in circuit. With the values shown, the circuit below is a nominally 4.2v 'zener', 6x 0.7v.
Beware that its knee is only as relatively sharp as a diode, it has a diode's tempco, and relatively high leakage at low voltages. Use with a darlington transistor for lower leakage.
If you have a low power zener diode to hand, and for some reason need a higher power one, then replacing R2 with a zener results in a circuit which behaves like the zener with another 0.7v drop, but the dissipation capability of Q1.
It depends how complete you want the emulation to be. D1 is only there to emulate the forward conduction. Most applications of zeners in voltage references will not need that.
Note that the voltage is not constant above the conduction voltage, but then neither is that of a zener, both have an effective series resistance.
If what you have got is only BJTs and resistors, then you could try using a reverse-biased BE junction as a rudimentary Zener in this way:
Be aware that the "Zener" voltage is not well defined (around 5V-8V) and the current in the "Zener" must be kept at reasonable level for the part you are using. Moreover, using a BJTs like this will permamently degrade its performance, i.e. don't reuse the BJT for anything else, because it will its Hfe greatly reduced.
In other words, you have to experiment and be prepared to throw away some BJTs if you are unlucky with your attempts.
BJTs are sometimes used like this instead of true Zeners as protection devices, i.e. when performance as a Zener is not critical and they must operate only in "emergency" (see also this EE.SE question).
EDIT (Prompted by a comment by the OP on the question)
Note that if you require a sharp "Zener" knee, i.e. a very stable value for the output voltage of the voltage reference, this is conflicting with your need of using only resistors, capacitors and BJTs.
Stable and precise values of a reference voltage can be obtained only with (quality) Zener diodes or (for extra accuracy/stability) voltage regulator ICs (sometimes called voltage reference ICs when they are not meant to power a significant load).
For example see the TL431 voltage regulator ICs, which is meant also as an "universal" Zener replacement. And when I said "quality" Zeners I meant something like the LTZ1000 reference diode. But this latter cost a fortune and is used in high end instrumentation. Excerpts from the LTZ1000 datasheet:
Included on the chip is a subsurface zener reference, a heater resistor for temperature stabilization, and a temperature sensing transistor. External circuitry is used to set operating currents and to temperature stabilize the reference. This allows maximum flexibility and best long-term stability and noise.