This is pretty much what we do to apply small bias currents to our superconducting circuits. The signals are small (<1 uA), and the power is dissipated outside of the cryostat, so this method is very simple and effective. The voltage and resistors don’t even need to be that huge, ~10 MOhm or below, and correspondingly, <10 V.
It works. In most applications it would be wasteful of electrical power (and money). You have to generate this very large voltage that (it turns out) you don't really need.
On the other hand that circuit is very easy to understand and build and test.
For a design without feedback, and in an energetically inefficient way, maybe this can work too:
1. Determine what will be the maximum resistance of the "current consumer" part of your circuit throughout its operation.
2. Prepare a resistor several magnitudes larger than the resistance above.
3. Connect to the resistor above a (huge) voltage source so that the resulting current is the one you target for your current source
4. Put the "current consumer" part of your circuit in series with the large resistor.